title: Coase theorem

text: Because Ronald Coase did not originally intend to set forth any one particular theorem, it has largely been the effort of others who have developed the loose formulation of the Coase theorem. What Coase initially provided was fuel in the form of “counterintuitive insight” that externalities necessarily involved more than a single party engaged in conflicting activities and must be treated as a reciprocal problem. His work explored the relationship between the parties and their conflicting activities and the role of assigned rights/liabilities. While the exact definition of the Coase theorem remains unsettled, there are two issues or claims within the theorem: the results will be efficient and the results in terms of resource allocation will be the same regardless of initial assignments of rights/liabilities.


title: Coase theorem

text: The zero transaction cost condition is taken to mean that there are no impediments to bargaining. Since any inefficient allocation leaves unexploited contractual opportunities, the allocation cannot be a contractual equilibrium.


title: Coase theorem

text: This version fits the legal cases cited by Coase. If it is more efficient to prevent cattle trampling a farmer's fields by fencing in the farm, rather than fencing in the cattle, the outcome of bargaining will be the fence around the farmer's fields, regardless of whether victim rights or unrestricted grazing-rights prevail. Subsequent authors have shown that this version of the theorem is not generally true, however. Changing liability placement changes wealth distribution, which in turn affects demand and prices.


title: Coase theorem

text: In his UCLA dissertation and in subsequent work, Steven N. S. Cheung (1969) coined an extension of the Coase theorem: aside from transaction costs, all institutional forms are capable of achieving the same efficient allocation. Contracts, extended markets, and corrective taxation are equally capable of internalizing an externality. To be logically correct, some restrictive assumptions are needed. First, spillover effects must be bilateral. This applies to the cases that Coase investigated. Cattle trample a farmer's fields; a building blocks sunlight to a neighbor's swimming pool; a confectioner disturbs a dentist's patients etc. In each case the source of the externality is matched with a particular victim. It does not apply to pollution generally, since there are typically multiple victims. Equivalence also requires that each institution has equivalent property rights. Victim rights in contract law correspond to victim entitlements in extended markets and to the polluter pays principle in taxation.


title: Coase theorem

text: Notwithstanding these restrictive assumptions, the equivalence version helps to underscore the Pigouvian fallacies that motivated Coase. Pigouvian taxation is revealed as not the only way to internalize an externality. Market and contractual institutions should also be considered, as well as corrective subsidies. The equivalence theorem also is a springboard for Coase's primary achievement—providing the pillars for the New Institutional Economics. First, the Coasean maximum-value solution becomes a benchmark by which institutions can be compared. And the institutional equivalence result establishes the motive for comparative institutional analysis and suggests the means by which institutions can be compared (according to their respective abilities to economize on transaction costs). The equivalency result also underlies Coase's (1937) proposition that the boundaries of the firm are chosen to minimize transaction costs. Aside from the "marketing costs" of using outside suppliers and the agency costs of central direction inside the firm, whether to put Fisher Body inside or outside of General Motors would have been a matter of indifference.


title: Coase theorem

text: The Coase Theorem has been used by jurists and legal scholars in the analysis and resolution of disputes involving both contract law and tort law.


title: Coase theorem

text: In contract law, the Coase theorem is often used as a method to evaluate the relative power of the parties during the negotiation and acceptance of a traditional or classical bargained-for contract.


title: Coase theorem

text: In modern tort law, application of economic analysis to assign liability for damages was popularized by Judge Learned Hand of the Second Circuit Court of Appeals in his decision, "United States v. Carroll Towing Co." 159 F.2d 169 (2d. Cir. 1947). Judge Hand's holding resolved simply that liability could be determined by applying the formula of formula_1, where formula_2 is the burden (economic or otherwise) of adequate protection against foreseeable damages, formula_3 is the probability of damage (or loss) occurring and formula_4 is the gravity of the resulting injury (loss). This decision flung open the doors of economic analysis in tort cases, thanks in no small part to Judge Hand's popularity among legal scholars.


title: Coase theorem

text: In resultant scholarship using economic models of analysis, prominently including the Coase theorem, theoretical models demonstrated that, when transaction costs are minimized or nonexistent, the legal appropriation of liability diminishes in importance or disappears completely. In other words, parties will arrive at an economically efficient solution that may ignore the legal framework in place.


title: Coase theorem

text: Two property owners own land on a mountainside. Property Owner A's land is upstream from Owner B and there is significant, damaging runoff from Owner A's land to Owner B's land. Four scenarios are considered:


title: Coase theorem

text: The Coase theorem considers all four of these outcomes logical because the economic incentives will be stronger than legal incentives. Pure or traditional legal analysis will expect that the wall will exist in both scenarios where B has a cause of action and that the wall will never exist if B has no cause of action.


title: Coase theorem

text: The Jones family plants pear trees on their property which is adjacent to the Smith family. The Smith family gets an external benefit from the Jones family's pear trees because they pick up whatever pears fall to the ground on their side of the property line. This is an externality because the Smith family does not pay the Jones family for utility received from gathering the fallen pears and, therefore, does not participate in the market transaction of pear production. It results in the pears being underproduced, which means too few pear trees are planted.


title: Coase theorem

text: By applying the Coase Theorem two possible solutions arise for internalizing this externality. These solutions can occur because the positive external benefits are clearly identified and we assume that 1) transaction costs are low; 2) property rights are clearly defined.


title: Coase theorem

text: After realizing that the Smith family gets utility from the Jones family's pear trees, the Jones family thinks it is unfair that the Smiths get utility from picking up the pears that fall from their pear trees. The first option to eliminate the externality could be to put up a net fence that will prevent pears from falling to the ground of the Smith's side property line, which will automatically decrease the Smith family's marginal benefit to 0.


title: Coase theorem

text: The second option for the Jones could be to impose a cost on the Smith family if they want to continue to get utility from their pear trees. Say, if the Jones family has a MC of $25 for each pear tree produced, it allows them to plant 3 pear trees a year (Jones’ MB = MC). However, if the cost is imposed on the Smiths, the optimal quantity of pear trees produced a year will increase to 4 (Jones’ MB + Smiths’ MB = MC). By internalizing the externality, both the Smith family and the Jones family increase their overall utility by increasing production from 3 pear trees a year to 4. $5 is the maximum price the Smiths are willing to pay for an additional, fourth, pear tree, which implies their marginal benefit to plant a fifth pear tree is 0.


title: Coase theorem

text: Whilst the Coase Theorem remains largely theoretical, there are some real life examples of Coasean bargaining. One Coasean example was the negotiations that took place between waterworks and farmers in Denmark. Danish waterworks were attempting to establish “voluntary cultivation agreements with Danish farmers”. Some main takeaways from this application of the Coase Theorem were that, the farmers tried to receive over compensation by exploiting their information advantage, which in turn may have resulted in waterworks terminating negotiations. Additionally, as asymmetric information is included in transaction costs, by exploiting their information advantage, the farmers prolonged negotiations. Thus demonstrating that the Coase theorem is very sensitive to its assumption of low transaction costs.


title: Coase theorem

text: While most critics find fault with the applicability of the Coase Theorem, a critique of the theorem itself can be found in the work of the critical legal scholar Duncan Kennedy, who argues that the initial allocation always matters in reality. This is because psychological studies indicate that asking prices often exceed offer prices, due to the so-called endowment effect. Essentially, a person who already has an entitlement is likely to request more to give it up than would a person who started off without the entitlement. The validity of this theoretical critique in practice is addressed in a later section.


title: Coase theorem

text: An additional critique of the theorem comes from new institutional economist Steven N. S. Cheung who thinks that private property rights are institutions that arise to reduce transaction costs. The existence of private property rights implies that transaction costs are non-zero. If transaction costs are really zero, any property rights system will result in identical and efficient resource allocation, and the assumption of private property rights is not necessary. Therefore, zero transaction costs and private property rights cannot logically coexist.


title: Coase theorem

text: Lastly, using a game-theoretic model, it has been argued that sometimes it is easier to reach an agreement if the initial property rights are unclear.


title: Coase theorem

text: Ronald Coase's work itself emphasized a problem in applying the Coase theorem: transactions are "often extremely costly, sufficiently costly at any rate to prevent many transactions that would be carried out in a world in which the pricing system worked without cost." (Coase, 1960—first paragraph of section VI.) This isn't a criticism of the theorem itself, since the theorem considers only those situations in which there are no transaction costs. Instead, it is an objection to applications of the theorem that neglect this crucial assumption.


title: Coase theorem

text: So, a key criticism is that the theorem is almost always inapplicable in economic reality, because real-world transaction costs are rarely low enough to allow for efficient bargaining. (That was the conclusion of Coase's original paper, making him the first 'critic' of using the theorem as a practical solution.) Neo-Keynesian economist James Meade argued that even in a simple case of a beekeeper's bees pollinating a nearby farmer's crops, Coasean bargaining is inefficient (though beekeepers and farmers do make contracts and have for some time).


title: Coase theorem

text: Chicago school anarcho-capitalist economist David D. Friedman has argued that the fact that an "economist as distinguished as Meade assumed an externality problem was insoluble save for government intervention suggests...the range of problems to which the Coasean solution is relevant may be greater than many would at first guess."


title: Coase theorem

text: In many cases of externalities, the parties might be a single large factory versus a thousand landowners nearby. In such situations, say the critics, the transaction costs rise extraordinarily high due to the fundamental difficulties in bargaining with a large number of individuals.


title: Coase theorem

text: However, transaction costs are not only a barrier for situations with large numbers of involved parties. Even in the simplest of situations, with only two individuals, social costs can increase transaction costs to be unreasonably high so as to invalidate the applicability of Coasean bargaining. As economist Jonathan Gruber describes, there are strong social norms that often prevent people from bargaining in most day-to-day situations. Whether it is the awkwardness of the exchange or the fear of greatly under-valuing someone else's property rights, transaction costs can still be quite high even in the most basic interactions that could make use of the Coase Theorem.


title: Coase theorem

text: Gruber further describes three additional issues with attempting to apply the Coase Theorem to real-world situations. The first of these is known as the assignment problem, and stems from the fact that for most situations with externalities, it is extremely difficult to determine who may be responsible for the externality as well as who is actually affected by it. Take the case of a polluted river that reduces the fish population. How can the involved parties determine which factories may have contributed the pollution that specifically harmed the fish, or whether there were any natural factors that interfered in the process. And even if we can determine who exactly is responsible and who is harmed, it is incredibly cumbersome to accurately quantify these effects. People cannot easily translate their experiences to monetary values, and they are likely to overstate the harm they have suffered. At the same time, the polluters would likely underestimate the proportion of the externality that they are responsible for.


title: Coase theorem

text: Second, in situations where multiple parties hold the property rights, Coasean bargaining often fails because of the holdout problem. Once all the property owners except for one have accepted the Coasean solution, the last party is able to demand more compensation from the opposing party in order to part with the property right. Knowing this, the other property owners have the incentive to also demand more, leading to the unraveling of the bargaining process.


title: Coase theorem

text: Lastly, if the side with only one party holds the property rights (so as to avoid the holdout problem), Coasean bargaining still fails because of the free-rider problem. When the multiple parties on the other side all benefit fairly equally from the results of the negotiations, then each of the parties has the incentive to free-ride, to withhold their payments and withdraw from the negotiations because they can still receive the benefits regardless of whether or not they contribute financially. Ellingsen and Paltseva (2016) model contract negotiation games and show that the only way to avoid the free-rider problem in situations with multiple parties is to enforce mandatory participation (such as through the use of court orders).


title: Coase theorem

text: In 2009, in their seminal JEI article, Hahnel and Sheeran highlight several major misinterpretations and common assumptions, which when accounted for substantially reduce the applicability of Coase's theorem to real world policy and economic problems. First, they recognize that the solution between a single polluter and single victim is a negotiation—not a market. As such, it is subject to the extensive work on bargaining games, negotiation, and game theory (specifically a "divide the pie" game under incomplete information). This typically yields a broad range of potential negotiated solutions, making it unlikely that the efficient outcome will be the one selected. Rather it is more likely to be determined by a host of factors including the structure of the negotiations, discount rates and other factors of relative bargaining strength (cf. Ariel Rubenstein).


title: Coase theorem

text: If the negotiation is not a single shot game, then reputation effects may also occur, which can dramatically distort outcomes and may even lead to failed negotiation (cf. David M. Kreps, also the chainstore paradox). Second, the information assumptions required to apply Coase's theorem correctly to yield an efficient result are complete information—in other words that both sides lack private information, that their true costs are completely known not only to themselves but to each other, and that this knowledge state is also common knowledge. When this is not the case, Coasean solutions predictably yield highly inefficient results because of perverse incentives—not "mere" transaction costs.


title: Coase theorem

text: If the polluter has the ownership rights, it is incentivized to overstate its benefits from polluting, if the victim has the ownership rights, (s)he has the incentive to also misrepresent her/his damages. As a result, under incomplete information (probably the only state of knowledge for most real world negotiations), Coasean bargaining yields predictably inefficient results.


title: Coase theorem

text: Hahnel and Sheeran emphasize that these failures are not due to behavioral issues or irrationality (although these are quite prevalent (ultimatum game, cognitive biases)), are not due to transaction costs (although these are also quite prevalent), and are not due to absorbing states and inability to pay. Rather, they are due to fundamental theoretical requirements of Coase's theorem (necessary conditions) that are typically grossly misunderstood, and that when not present systematically eliminate the ability of Coaseian approaches to obtain efficient outcomes—locking in inefficient ones. Hahnel and Sheeran conclude that it is highly unlikely that conditions required for an efficient Coaseian solution will exist in any real-world economic situations.


title: Coase theorem

text: Unconstrained Coasean bargaining ex post may also lead to a hold-up problem ex ante. Thus, even though it is often claimed that Coasean bargaining is an alternative to Pigouvian taxation, it has been argued that in a hold-up situation Coasean bargaining may actually justify a Pigouvian tax. Alternatively, it may be efficient to forbid renegotiation. Yet, there are situations in which a ban on contract renegotiation is not desirable.


title: Coase theorem

text: Unlike Hahnel and Sheeran, the economist Richard Thaler highlights the importance of behavioral economics in explaining the inability to effectively use the Coase Theorem in practice. Thaler modified his experiments with the Ultimatum game and showed that people were more likely to be concerned with ensuring fairness in negotiations when negotiating over their own tangible property rather than in an abstract sense. This suggests that in practice, people would not be willing to accept the efficient outcomes prescribed by the Coasean bargaining if they deem them to be unfair. So, while the Coase theorem suggests that parties who lose out on property rights should then pursue the property according to how much they value it, this does not often happen in reality. For example, Professor Ward Farnsworth has described how in the aftermath of twenty observed legal nuisance cases, none of the parties ever attempted to engage in Coasean bargaining (as would be expected to reach the most efficient outcome) because of anger at the unfairness of having to bargain. It is possible that Coase and his defenders would simply view this as a non-pecuniary transaction cost, but that may be an unreasonable extension of the concept of transaction costs.


title: Coase theorem

text: Thaler has also provided experimental evidence for the argument that initial allocations matter, put forth by Duncan Kennedy (as previously noted), among others. When students were trading cash-equivalent tokens, the negotiations resulted in the students who would receive the most cash from a token (as told by the researchers) holding the tokens, as would be predicted by the Coase Theorem. However, when the students were trading property (mugs in this case) that were not directly equivalent to cash, proper Coasean bargaining did not occur as depicted in the adjacent diagram. This is because people generally exhibit an endowment effect, in which they value something more once they actually have possession of it. Thus, the Coase Theorem would not always work in practice because initial allocations of property rights would affect the result of the negotiations.


title: Coase theorem

text: Ian A. MacKenzie and Markus Ohndorf have conducted research on Coasean bargaining in the presence of a Pigouvian tax. This research stems from the common belief within Coasean perspectives that Pigouvian taxation creates distortions and therefore inefficiencies, instead of effectively resolving the problem in question. The research conducted shows that in the presence of a pre-existing Pigouvian tax, Coasean bargaining may be superior. The implications of this policy are regulation at both the federal and state level and environmental litigation and liability. This is because dual regulatory environments exist.


title: Coase theorem

text: In order to examine whether the hypothesis that Coasean bargaining in the presence of a Pigouvian tax is superior to a scenario without taxation, MacKenzie and Ohndorf had to make certain assumptions. Firstly, they relaxing the assumption of property right allocations, and in doing so had a new take on the Buchanan – Stubblebine – Turvey Theorem. By relaxing this assumption, they are able to conclude that even with a Pigouvian tax levied, efficiency improvements can exist. By creating a more realistic environment with how property rights are allocated, MacKenzie and Ohndorf observed that gains from Coasean exchange are reduced by a Pigouvian tax. Furthermore, their research also shows that it is possible that a pre-existing tax can be efficiency enhancing in the case of environmental litigation and liability. This is because it softens the controversy and therefore, reduces overall spending in litigation.


title: Coase theorem

text: Therefore, in summary, MacKenzie and Ohndorf's research provides an economic argument in support of Pigouvian taxation in the case where there is the potential for negotiation.


title: Tughra

text: A tughra () is a calligraphic monogram, seal or signature of a sultan that was affixed to all official documents and correspondence. Inspired by the tamgha, it was also carved on his seal and stamped on the coins minted during his reign. Very elaborate decorated versions were created for important documents that were also works of art in the tradition of Ottoman illumination, such as the example of Suleiman the Magnificent in the gallery below.


title: Tughra

text: The tughra was designed at the beginning of the sultan's reign and drawn by the court calligrapher or "nişancı" on written documents. The first tughra examples are from the 14th century.


title: Tughra

text: Tughras served a purpose similar to the cartouche in ancient Egypt or the Royal Cypher of British monarchs. Every Ottoman sultan had his own individual tughra.


title: Tughra

text: There are two main schools of thought on the origins of the word tughra. The first sees it derived from a Turkic secretarial emblem called "tughragh", and the second as an effort by Persian scribes to shape the name of the ruler into a bow-like element called "turgha"/"turghay", subsequently mispronounced as tughra.


title: Tughra

text: The primary argument for the first school is a remark by Mahmud al-Kashgari in his "Dīwān Lughāt al-Turk":


title: Tughra

text: The tughra has a characteristic form, two loops on the left side, three vertical lines in the middle, stacked writing on the bottom and two extensions to the right. Each of these elements has a specific meaning, and together they make up the form that is easily recognizable as a tughra.


title: Tughra

text: The name of the sultan is written out in the bottom section, called a "sere". Depending on the period, this name can be as simple as Orhan, son of Osman, in the first tughra in 1326. In later periods honorifics and prayers are also added to the name of the tughra holder and his father.


title: Tughra

text: The loops to the left of the tughra are called "beyze", from Arabic meaning "egg". Some interpretations of tughra design claim that the beyzes are supposed to symbolize the two seas the sultans held sway over: the outer larger loop signifying the Mediterranean and the inner, smaller loop signifying the Black Sea.


title: Tughra

text: The vertical lines on the top of the tughra are called "tuğ", or flagstaff. The three tugs signify independence. The S-shaped lines crossing the tugs are called "zülfe" and they, together with the tops of the tugs that also look to the right, signify that the winds blow from the east to the west, the traditional movement of the Ottomans.


title: Tughra

text: The lines to the right of the tughra are called "hançer" and signify a sword, symbol of power and might.


title: Tughra

text: Although the tughra is largely identified with the Ottoman Sultans, they have also sometimes been used in other states, such as the Qajar dynasty, Safavid Empire and the Khanate of Kazan. Later, tughras were used among the Tatars of Imperial Russia.


title: Tughra

text: The Mughal Emperors are also known to have used calligraphic symbols, alongside the Ottomans, the Mughal "Tughra" was circular in shape with three points at its tip, beside the calligraphic signature of the emperor.


title: Tughra

text: Afghan currency notes from 1919 to 1936 had the tughra present as well. Pakistan had the tughra on it coins from 1947 till 1974; both of these are present in the State Bank Museum in Karachi. The nawab of Bahawalpur and the Nizam of Hyderabad had tugras on their coinage as well. The flowing lines could symbolize the wide reach of Suleyman's rule and his future conquests. It could also signify the spread of Islam to other realms beyond the Ottoman Empire.


title: Tughra

text: There are modern artists of calligraphy that use the characteristic tughra form today. Examples are the tughras of Russian president Vladimir Putin and Akihito, the Emperor of Japan, created by artist Vladimir Popov.


title: Lydia (The Walking Dead)

text: Lydia is a fictional character from the comic book series "The Walking Dead" and the television series of the same name, where she is portrayed by Cassady McClincy.


title: Lydia (The Walking Dead)

text: Lydia is one of the Whisperers to follow Paul "Jesus" Monroe, Darius and two other guards. She is captured by Jesus. She is put in a cell next to Carl's. She tries to get information about the Hilltop community by talking to Carl, and later starts a sexual relationship with him. After the war against his group, Lydia goes to the ranks of Rick's group and survives to the end of the story. Carl's lingering feelings for Sophia finally ends their relationship, although they remain close friends.


title: Lydia (The Walking Dead)

text: Lydia is a survivor who, along with her parents, "Alpha" and Frank, survived the initial stages of the zombie apocalypse. After 23 days of the initial outbreak, Lydia and her parents remained in a moldy basement with a group of survivors. After Frank dies, Alpha raises Lydia with cruelty, treating her more like her servant than her child, to the point of making Lydia call her "Alpha" instead of "momma". After being captured by Daryl Dixon and Michonne, and kept as a prisoner in a cell at the Hilltop, Lydia begins to form a bond with Henry. Lydia and Henry fall in love and begin a relationship, with her ultimately rejecting the Whisperers in favor of the other communities who agree to grant Lydia asylum. Lydia rejects the Whisperers completely and is disowned by her mother. However, her relationship with Henry is cut short when Alpha brutally murders him and several other residents of Hilltop.


title: Lydia (The Walking Dead)

text: In the episode "Dee," taking place between the flashback scenes of "Omega" and "We Are the End of the World," Lydia and her mother live on a steamboat run by a woman named Brooke. Lydia is nearing her ninth birthday and bonds with Brooke while remaining fearful of her mother, currently going by her real name of Dee, due to Dee's murder of Lydia's father Frank. Lydia enjoys the community's efforts to recapture the pre-apocalypse world, although Dee sees Brooke as a bad influence on her daughter and has a much more realistic view of the world, although at this time, Dee has been trying to be a good person and mother rather than being as ruthless as she usually is as Alpha. Dee's violent reaction to a suspicious bartender named Billy sends Lydia into hiding when it stirs up bad memories for her. However, Dee proves to be right about Billy who returns to take the boat for himself and demands that Dee and Lydia be handed over. Dee manages to escape to shore with Lydia while a fight breaks out that decimates both sides.


title: Lydia (The Walking Dead)

text: On shore, Dee and Lydia narrowly avoid a herd that devours the survivors by hiding under a corpse. After they discover that Brooke is the only other survivor, Lydia stops her mother from killing Brooke and remains defiant towards Dee. Lydia refuses to learn how to fight a herd and runs off, causing Dee to realize her mistakes and that this is no world for a child. As Dee prepares to kill Lydia and then herself, they are surrounded by the Whisperers, currently led by a woman named Hera. Hera and the Whisperers take in Dee and Lydia with Dee eventually killing Hera and becoming Alpha, the leader of the Whisperers.


title: Lydia (The Walking Dead)

text: In the mid-season premiere, "Adaptation", as Daryl, Michonne, and the others return to the Hilltop, they spot six walkers following them and lure the group into a confrontation on a covered bridge. Three of the "walkers" turn out to be humans in disguise. Daryl and Michonne kill two while the third—Lydia—surrenders and is taken prisoner back to the Hilltop. Lydia is locked up and questioned by Daryl. After Daryl leaves, Lydia introduces herself to Henry and the two start to get to know each other; Daryl overhears this conversation. In the episode "Omega", Lydia describes her backstory to Henry while both are in the community prison, unaware Daryl is listening in.


title: Lydia (The Walking Dead)

text: As Lydia explains, Henry starts to become more compassionate towards Lydia. Daryl continues to talk to her as well, trying to scope out the size of her group as Lydia claims that they may be coming to save her. He finds her arm covered in marks, which she says came from her mother. Daryl, who had been similarly beaten as a child, displays sympathy for Lydia's abuse, but she remains uncooperative. Lydia's story is told in flashbacks that date to the months after the walker outbreak. Henry decides to let Lydia out of her cell for a while and discreetly shows her what the Hilltop has to offer. While Lydia has an opportunity to take a hammer and strike Henry with it to escape, the cries of a baby cause her to panic, and she begs to be returned to her cell. She asks Henry to stay with her through the night. When morning comes, Daryl returns and Lydia states that she doesn't think her mother will be coming. Lydia then tells them that she had lied about her history. Lydia mentions where her mother and her people may be camped, but notes that they frequently move about.


title: Lydia (The Walking Dead)

text: In the episode "Bounty", Lydia's mother, Alpha, arrives at Hilltop with some of the Whisperers and demands the release of Lydia. Inside the Hilltop, Daryl discovers that Henry and Lydia have escaped. Enid and Addy are able to track down Henry and Lydia to an abandoned house near the Hilltop, where Henry has provided Lydia a new set of clothes. Lydia opts to return to Hilltop of her own accord and is returned to Alpha in exchange for the return of Alden and Luke, two Hilltop residents being held prisoner by the Whisperers. Reunited, Lydia apologizes to her mother. Alpha slaps her across the face, ordering her to address her as "Alpha" like everyone else does.


title: Lydia (The Walking Dead)

text: In the episode "Guardians", as Alpha leads her group back to camp, she questions Lydia about her time at the Hilltop, but Lydia claims that she has little information of interest. Henry eventually catches up with Lydia and the Whisperers, but is found and captured by Beta, Alpha's second-in-command. Henry reveals that he came alone to rescue Lydia. Alpha decides that Henry is coming with them. On their way back to camp, Alpha asks Lydia why she didn't mention Henry. Lydia explains that he wasn't worth mentioning, but Alpha reminds her that he just risked his life to save her, so she must be lying. The Whisperers soon arrive back at their camp. At night, Beta takes Henry to Alpha. Unmasked, Alpha drops her knife and makes Lydia pick it up, commanding her to kill Henry with it so that she can prove what side she's on; Lydia picks up the knife and begins to cry. Her mother warns her not to be weak and that Beta will kill them both if she doesn't kill Henry. Suddenly, a small horde of walkers show up and begin eating the unmasked Whisperers, creating mass confusion. Alpha and the others quickly put on their masks so that they can try to lure the herd away. Moments later, Daryl and Connie arrive disguised as Whisperers to free and rescue Henry; they prepare to retreat. Daryl grabs Henry, who grabs Lydia, and together the four escape.


title: Lydia (The Walking Dead)

text: In the episode "Scars", Daryl, Connie, Henry and Lydia take shelter at Alexandria. That evening, Daryl, Connie, Henry and Lydia depart for the Kingdom.


title: Lydia (The Walking Dead)

text: In the episode "The Calm Before", after the fair at the Kingdom begins, Alpha sneaks into the fair by posing as an Alexandria resident. That night, as most of the fairgoers gather to watch a movie, Lydia is saving a seat for Henry when Alpha quietly sits next to her and gestures for her to stay quiet. Outside the theater, Alpha tries to convince Lydia to leave the Kingdom with her, but Lydia rejects her. Alpha tells her she is not strong and is no longer part of her group, and departs. Later, Daryl's group finds a line of ten decapitated heads on pikes, one of which is Henry's. Sometime later, Daryl and Lydia return to the border. Lydia leaves the necklace Henry made for her on the ground by the pike that once held Henry's head.


title: Lydia (The Walking Dead)

text: In the season finale "The Storm", Ezekiel makes the decision to abandon the Kingdom, assembling a large caravan of the remaining survivors to seek out better shelter at the Hilltop. Lydia wanders into the forest and finds a walker half-buried in the snow. She takes off her glove while crying and offers her hand to the walker to commit suicide, but stops when Carol appears and follows her back to the group. Using the cover of the storm, the Kingdom's group passes the pikes into Alpha's territory, finding only frozen walkers along the way, and arrive at a river that has been frozen over. Carol goes looking for Lydia, who has disappeared. She eventually catches up to Lydia, who believes that she is to blame for their situation and feels she can't stay with them. Since she cannot go back to her mother, Lydia asks Carol to kill her, but Carol refuses Lydia's request and tells her that she is not weak.


title: Lydia (The Walking Dead)

text: Although now a resident of Alexandria, Lydia is often distrusted and bullied, particularly by the other teenagers of the community who lost their friends in Alpha's massacre despite Lydia's protestations that she had truly loved Henry. Lydia bonds with Negan over them both being outcasts and Negan rescues Lydia from an attack by her vengeful bullies, accidentally killing one of them in the process, but also putting Negan's own life on the line. Lydia's testimony proves to be enough to convince Daryl to spare Negan, but he escapes before a final decision can be made and joins the Whisperers. Daryl continues to look after Lydia and act as a father figure towards her while Michonne in particular seeks to keep her safe, knowing that Lydia's presence may cause Alpha to hesitate to attack.


title: Lydia (The Walking Dead)

text: During this time, Lydia helps to defend the communities, having taken up Henry's old fighting staff as her weapon and adopting the fighting style that he had learned from Morgan Jones. Carol's attempts to manipulate Gamma by revealing Lydia's survival to her causes Lydia to abandon the communities and take off on her own, but she later returns, although she can't bring herself to kill her mother. During the Whisperer attack on the Hilltop, Lydia joins in the defense and suffers an injured leg. She is later captured by Negan who leads Alpha to where he is supposedly keeping Lydia with Alpha planning to kill her daughter to that she will rejoin the Whisperers as a walker. However, this turns out to be a trap and Negan kills Alpha. Lydia manages to escape before Negan can return and alerts Daryl who confronts Negan, learning that Negan was working with Carol to infiltrate the Whisperers in order to assassinate Alpha and had used Lydia to draw her out.


title: Lydia (The Walking Dead)

text: In the aftermath of her mother's death, Lydia is left with conflicted feelings as she truly loved Alpha despite everything. When the Coalition's base the Tower comes under siege by Beta and his massive horde, Lydia helps direct the Coalition's archers in providing covering fire against the Whisperers, picking a few Whisperers out of the horde for them. Lydia urges Negan to help and he provides her with Alpha's Whisperer mask. Using the mask, Lydia infiltrates the horde and joins Carol, Daryl and the rest of their team in diverting the horde. When the Whisperers destroy their makeshift sound system, Lydia offers to lead the horde away herself, having been trained in how to do it by her mother despite the fact that it is very likely to be a suicide mission. As the others assassinate the Whisperers one by one, Lydia diverts the herd and is saved from retaliation by Beta by Negan and Daryl who finally take Beta down. With the Whisperers defeated, Lydia leads the horde to a massive cliff where Carol takes over, intending to sacrifice herself to finish the job. Lydia saves Carol's life and together the two watch the destruction of the Whisperers' ultimate weapon. As the horde is destroyed, Lydia tosses her mother's mask over the cliff as well. Returning to the others, Carol hails Lydia as the one responsible for the end of the horde's threat and Negan tells Lydia that he intends to stick around for now.


title: Lydia (The Walking Dead)

text: After the defeat of the Whisperers, Lydia can briefly be seen helping the wagons carrying the survivors back home and helping to rebuild the badly damaged Alexandria.


title: Lydia (The Walking Dead)

text: With Alexandria badly damaged by the Whisperers and their food supply destroyed, Lydia, now a fully accepted and respected member of the community, joins the efforts to gather replacement food supplies, participating in the mission to a military base. When a team returns to the ruins of the Hilltop in search of blacksmithing supplies to try to fix Alexandria's damaged wall, Lydia joins them and they discover a small faction of surviving Whisperers squatting in the ruins. Lydia recognizes their leader, Keith, as a Whisperer who was kind to her behind Alpha's back and not a part of her mother's inner circle. Lydia is horrified by the barbarity that her vengeful friends show to Keith, but Keith proves to be more understanding. Later, when a herd breaches Alexandria during a massive storm, Lydia participates in defending the survivors taking refuge inside of a house. When Aaron falls into trouble, Lydia escapes through a window and comes to his rescue. When the team who went to Meridian returns, Lydia is visibly disappointed and upset by Negan's choice to take off on his own.


title: Lydia (The Walking Dead)

text: After the communities make contact with the Commonwealth, Lydia moves to the Hilltop and joins in Maggie's efforts to rebuild the community. Maggie's young friend Elijah develops a crush on Lydia who considers leaving to join the Commonwealth. However, the arrival of a dying young man leads Maggie, Lydia and Elijah to the community of Riverbend which is under siege by the Commonwealth Army with Gabriel and Aaron caught in the middle. Lydia is reunited with Negan who now lives in Riverbend with his pregnant wife Annie. Lydia and the others help the Riverbend residents to defeat the invaders and Lydia helps them to hide in an old Whisperer hideout in the aftermath and then helps Maggie, Elijah and Marco to fight off an invasion of the Hilltop by Commonwealth soldiers and Leah Shaw. When Lance Hornsby seizes control of Alexandria, the Hilltop and Oceanside, Lydia is amongst a group that remains outside of the Commonwealth's control to fight back.


title: Lydia (The Walking Dead)

text: After the defeat of Lance Hornsby, Lydia joins Aaron, Jerry and Elijah on a supply run to Oceanside. Lydia and Elijah continue developing feelings for each other, but Lydia resists the idea of a romantic relationship with him due to her lingering love for Henry. While stopping to avoid a herd, Aaron gives Lydia advice based upon his own romantic relationship with his late husband Eric Raleigh. Following an attack by a variant walker, Lydia finally accepts her feelings for Elijah and kisses him, starting a relationship with Elijah.


title: Lydia (The Walking Dead)

text: Shortly thereafter, the group encounters Luke and Jules who reveal that Oceanside has been taken over by Commonwealth forces. With Commonwealth soldiers searching the area, Lydia has the group cover themselves in walker guts and move with a nearby herd as a cover. However, the soldiers begin leading the herd towards the Commonwealth. Unbeknownst to everyone, the herd contains a number of variant walkers, one of which picks up a knife that Lydia drops. Trying to hide in an abandoned RV, the group is separated, and Lydia is bitten in the arm while trying to pull Elijah in. Jerry is forced to amputate Lydia's arm and leaves to find the others while Aaron remains behind with Lydia. Aaron and Lydia eventually manage to make their way through the herd to where it is overrunning the Commonwealth, only to learn that Luke and Jules are dead, and that Jerry and Elijah are still missing. Lydia receives medical treatment for her amputated arm and survives both it and the bite, but she is left despondent by Elijah's continued absence and the thought that someone she loves might once again be dead. After Pamela Milton's defeat, Lydia is finally reunited with Elijah who arrives with Jerry as the herd closes in on the part of the Commonwealth that most of the survivors are taking refuge in. Lydia joins the successful effort to destroy the herd and save the Commonwealth.


title: Lydia (The Walking Dead)

text: A year later, Lydia is living in the rebuilt Hilltop with Elijah whom she remains in a relationship with. Visiting Alexandria, Lydia delivers a package from Negan to Judith.


title: Lydia (The Walking Dead)

text: Lydia is portrayed by Cassady McClincy on "The Walking Dead" television series, beginning with the ninth season. She makes her first appearance in the episode "Adaptation".


title: Lydia (The Walking Dead)

text: Erik Kain of Forbes reviewed the episode "Chokepoint" and asserted that Lydia's relationship with Henry appeared "to have popped out of nowhere and blossomed much too quickly... it would have been more believable if their relationship had more time to grow and develop before Alpha came and took Lydia back".


title: Lydia (The Walking Dead)

text: Aaron Neuwirth of "We Live Entertainment" reviewed the episode "The Storm", stating, "Cassady McClincy has done a lot with a character who could have come off as the worst throughout these past few episodes that have heavily relied on her".


title: Lydia (The Walking Dead)

text: Alex McLevy of "The A.V. Club" noted that "Alpha’s daughter has been a surprisingly strong presence...and it has everything to do with Cassady McClincy’s performance. She has consistently located the raw-wound heart of the character, making even those groan-inducing moments of teenage romance between Henry and herself maintain a degree of appeal that would otherwise curdle into cloying nonsense".


title: Niagara River

text: The Niagara River () is a river that flows north from Lake Erie to Lake Ontario. It forms part of the border between the province of Ontario in Canada (on the west) and the state of New York in the United States (on the east). There are differing theories as to the origin of the river's name. According to Iroquoian scholar Bruce Trigger, "Niagara" is derived from the name given to a branch of the locally residing native Neutral Confederacy, who are described as being called the "Niagagarega" people on several late-17th-century French maps of the area. According to George R. Stewart, it comes from the name of an Iroquois town called "Ongniaahra", meaning "point of land cut in two".


title: Niagara River

text: The river, which is occasionally described as a strait, is about long and includes Niagara Falls in its course. The falls have moved approximately upstream from the Niagara Escarpment in the last 12,000 years, resulting in a gorge below the falls. Today, the diversion of the river for electrical generation has significantly reduced the rate of erosion. The total drop in elevation along the river is . The Niagara Gorge extends downstream from the Falls and includes the Niagara Whirlpool and another section of rapids.


title: Niagara River

text: Power plants on the river include the Sir Adam Beck Hydroelectric Power Stations (built in 1922 and 1954) on the Canadian side, and the Robert Moses Niagara Power Plant (built in 1961) on the American side. Together, they generate 4.4 gigawatts of electricity. The International Control Works, built in 1954, regulates the river flow. Ships on the Great Lakes use the Welland Canal, part of the Saint Lawrence Seaway, on the Canadian side of the river, to bypass Niagara Falls.


title: Niagara River

text: The Niagara River also features two large islands and numerous smaller islands. Grand Island and Navy Island, the two largest islands, are on the American and Canadian sides of the river, respectively. Goat Island and the tiny Luna Island split Niagara Falls into its three sections, the Horseshoe Falls, Bridal Veil Falls, and American Falls. Unity Island lies further upstream, alongside the city of Buffalo.


title: Niagara River

text: The Niagara River and its tributaries, Tonawanda Creek and the Welland River, formed part of the last section of the Erie Canal and Welland Canal. After leaving Lockport, New York, the Erie Canal proceeds southwest until it enters Tonawanda Creek. After entering the Niagara River, watercraft then proceed southward to the final lock, where a short section of the canal allows boats to avoid the turbulent shoal water at the river intake and enter Lake Erie. The Welland Canals used the Welland River as a connection to the Niagara River south of the falls, allowing water traffic to safely re-enter the Niagara River and proceed to Lake Erie.


title: Niagara River

text: The Niagara River and Falls have been known outside of North America since the late 17th century, when Father Louis Hennepin, a French explorer, first witnessed them. He wrote about his travels in "A New Discovery of a Vast Country in America" (1698).


title: Niagara River

text: The Niagara River was the site of the earliest recorded railway in America. It was an inclined wooden tramway built by John Montresor (1736–1799), a British military engineer, in 1764. Called "The Cradles" and "The Old Lewiston Incline," it featured loaded carts pulled up wooden rails by rope. It facilitated the movement of goods over the Niagara Escarpment in present-day Lewiston, New York.


title: Niagara River

text: In 1781, the Niagara Purchase was signed, involving in a six-and-a-half kilometre-wide strip of land bordering on the west bank of the Niagara River, connecting Lake Erie and Lake Ontario.


title: Niagara River

text: Several battles occurred along the Niagara River, which was historically defended by Fort George (Canadian side) and Fort Niagara (American side) at the mouth of the river and Fort Erie (Canadian side) at the head of the river. These forts were important during the Seven Years' War (known as the French and Indian War in the United States) and the American Revolutionary War. The Battle of Queenston Heights took place near the river in the War of 1812.


title: Niagara River

text: The river was an important route to liberation before the American Civil War, when many African-Americans escaping slavery on the Underground Railroad crossed it to find freedom in Canada. The Freedom Crossing Monument stands on the bank of the river in Lewiston to commemorate the courage of the escaping slaves and the local volunteers who helped them secretly cross the river.


title: Niagara River

text: In the 1880s, the Niagara River became the first waterway in North America harnessed for large-scale generation of hydroelectricity.


title: Niagara River

text: On the Canadian side of the river the provincial agency Niagara Parks Commission maintains all of the shoreline property, including Fort Erie, except the sites of Fort George (a National Historic Site maintained federally by Parks Canada), as a public greenspace and environmental heritage.


title: Niagara River

text: On the American side, the New York State Office of Parks, Recreation and Historic Preservation maintains several state parks adjacent to Niagara Falls and the Niagara River.


title: Niagara River

text: The Niagara River is listed as a Great Lakes Areas of Concern in The Great Lakes Water Quality Agreement between the United States and Canada.


title: Niagara River

text: The Niagara River has a long history of both road and rail bridges spanning the river, both upstream and downstream of the Falls. This history includes numerous bridges that have fallen victim to the harsh conditions of the Niagara Gorge, such as landslides and icepacks.


title: Niagara River

text: United States Coast Guard Fort Niagara Station was once a United States Army post. There are no Canadian Coast Guard posts along the river. Fort Mississauga, Fort George and Fort Erie are former British and Canadian military forts (last used 1953, 1965 and 1923 respectively) and are now parks.


title: Niagara River

text: Navy Island Royal Naval Shipyard was used by French in 18th Century as a naval base and by Royal Navy from 1763 as a small shipyard and abandoned around 1818 after the ratification of the Rush–Bagot Treaty in 1817.


title: Steve Ferrone

text: Steve Ferrone (born 25 April 1950) is an English drummer. He is known as a member of the rock band Tom Petty and the Heartbreakers from 1994 to 2017, replacing original drummer Stan Lynch, and as part of the "classic lineup" of the Average White Band in the 1970s. Ferrone has recorded and performed with Michael Jackson, Eric Clapton, George Harrison, Duran Duran, Stevie Nicks, Laura Pausini, Christine McVie, Rick James, Slash, Chaka Khan, Bee Gees, Scritti Politti, Aerosmith, Al Jarreau, Mick Jagger, Johnny Cash, Todd Rundgren and Pat Metheny. Ferrone also hosts 'The New Guy' weekly radio show on Sirius Xm's 'Tom Petty Radio'.


title: Steve Ferrone

text: Ferrone played with the band Bloodstone, appearing on their 1975 album "Riddle of the Sphinx". He then began playing with Brian Auger's band Oblivion Express, which had previously featured drummer Robbie McIntosh.


title: Steve Ferrone

text: McIntosh later joined the Average White Band and had just released their first number one album when McIntosh died of a heroin overdose. Ferrone was asked to join AWB in his place and stayed with them for the next eight years, recording and playing concert tours to support several hit albums, until AWB broke up in 1982.


title: Steve Ferrone

text: Since appearing on Chaka Khan's 1978 debut album, he went on to play on most of her subsequent albums in the 80s, including "Naughty" with former AWB band member Hamish Stuart. In 1985, Ferrone joined the "Saturday Night Live" house band and was there for the 1985–86 season. He contributed as session drummer for Duran Duran on the "Notorious", "Duran Duran", and "Thank You" albums. He also toured with Duran Duran on the Strange Behaviour Tour in support of "Notorious". He also toured and recorded with Eric Clapton from 1986 to 1992. During rehearsals in London for the George Harrison tour, he played drums for Duran Duran at Abbey Road Studios and recorded "Ordinary World." In 1991, Ferrone played drums on the George Harrison tour in Japan including three nights at the Tokyo Dome.


title: Steve Ferrone

text: Ferrone has also appeared on recordings by Eric Clapton, Bee Gees, Scritti Politti, Brian May, Anita Baker, George Benson, Jonathan Butler, Cyndi Lauper, Bryan Ferry, Climie Fisher, Christine McVie, Peter Frampton, Jeff Golub, Rick James, Whitney Houston, Chaka Khan, Freddie King, Tracy Chapman, Pat Metheny, Marcus Miller, Steve Winwood, Michael W. Smith, Morrissey–Mullen, Dick Morrissey, Jeffrey Osborne, Paul Simon, Alan Frew, Laura Pausini, Bernie Worrell, Al Jarreau, Michael Jackson and Jaco Pastorius.


title: Steve Ferrone

text: Ferrone substituted for Joey Kramer of Aerosmith during pre-production of the band's album "Nine Lives" while Kramer was grieving the loss of his father.


title: Steve Ferrone

text: Ferrone met guitarist Mike Campbell of Tom Petty and the Heartbreakers when the two played a show with George Harrison as part of his Hara and the Hijack band at Royal Albert Hall. He became the band's full-time drummer after the departure of original drummer Stan Lynch in 1994. Ferrone's first album with the band was Tom Petty's second solo outing "Wildflowers", released that same year. His first live performance as a member of the Heartbreakers was in Louisville, Kentucky on 28 February 1995.


title: Steve Ferrone

text: Ferrone thoroughly enjoyed his time with the Heartbreakers, saying that "everyone should have an experience like The Heartbreakers in their musical life: creativity, passion, honesty, integrity and a lot of fun."


title: Steve Ferrone

text: In November 2002, Ferrone founded Drumroll Studios in Burbank, California with financial advisor Alan Arora and the help and technical expertise of producer John Jones, with whom he had worked since Duran Duran's "Ordinary World" in 1992. Later, they were joined by guitarist and producer Steve Postell. Together they recorded and produced albums and tracks for a wide variety of musical acts, as well as film and TV scores.


title: Steve Ferrone

text: Ferrone released a live solo album titled "It Up: Steve Ferrone and Friends Live at La Ve Lee" in 2003.


title: Steve Ferrone

text: In 2007, he played drums on the self-titled debut album by The Black and White Years, produced by Jerry Harrison of Talking Heads.


title: Steve Ferrone

text: In 2010, Ferrone played on the eponymous first solo album by Slash, and joined the Italian band Pooh. He also worked on a collaborative project with composer and musician Eric Alexandrakis, guitarist and composer Warren Cuccurullo, and producer and songwriter Anthony J. Resta. Explaining the purpose of that project in a Modern Drummer news release, Alexandrakis said, "The four of us decided to create a scoring collective to pursue scoring projects in TV themes, film and advertising."


title: Steve Ferrone

text: In August 2018 he was announced as drummer for Dire Straits Legacy prior to their first US tour, consisting of 10 dates.


title: Steve Ferrone

text: In 2011 Ferrone began playing with old friends Alan Clark and Phil Palmer in a band made up of Dire Straits alumni called The Straits. The band also consisted of Mickey Feat on bass, Chris White on saxophone, Jaimie Squire on keyboards and vocals, and Terrence Reiss on vocals and guitar.


title: Steve Ferrone

text: In July 2015, Ferrone reunited with his AWB bandmates Malcolm "Molly" Duncan and Hamish Stuart to form The 360 Band. This is in essence one half of the classic Average White Band lineup. They released an album titled "Three Sixty" in 2017 and have been performing live together along with supporting musicians.


title: Steve Ferrone

text: In 2018, Ferrone played in the house band at a Porsche 70th anniversary event alongside fellow Heartbreaker Benmont Tench, as well as Ian Astbury, Dhani Harrison, and Brian Bell of Weezer


title: Steve Ferrone

text: On 10 July 2019, Ferrone was inducted into the Brighton Music Walk of Fame in his hometown of Brighton, England.


title: Steve Ferrone

text: On 1 Feb 2022, it was announced that Ferrone would be a part of the band for John Mayer's Sob Rock Tour.


title: Karolina Wydra

text: Karolina Wydra ( ; born March 5, 1981) is a Polish-American actress and model. She played Dominika Petrova on the Fox medical drama series "House", and vampire Violet Mazurski on the HBO dark fantasy series "True Blood". Wydra has starred in the fantasy thriller "After" (2012), and the science fiction film "Europa Report" (2013). She portrayed Detective Dianne Kubek on ABC's short-lived crime drama series "Wicked City" and recurred as Izel in "Agents of S.H.I.E.L.D."


title: Karolina Wydra

text: Wydra was born in Opole, Poland with a rare left eye coloboma birth defect in which the iris of the eye does not grow closed during gestation.


title: Karolina Wydra

text: Her mother and father were teachers of mathematics and art, respectively. In 1992, she and her family emigrated to Orange County, California, where her parents set up a cleaning business. Her parents separated in 2012, and her mother subsequently moved back to Poland.


title: Karolina Wydra

text: In October 1997, Wydra won the Elite Lee Jeans Model Look Contest, which was hosted by Roshumba Williams and Dorian Gregory. She has since appeared in print advertisements for brands such as Armani Exchange, Levi's Red, Calvin Klein, Smashbox Cosmetics, Dooney & Bourke, Charles David, Urban Decay, John Frieda, and Kenneth Cole. Wydra has also appeared as a model on the cover of the German "Elle" magazine and "Oyster" magazine.


title: Karolina Wydra

text: In 2006, Wydra starred in a Nespresso commercial with George Clooney. In 2008, she co-starred in the comedy-drama film "Be Kind Rewind", directed by Michel Gondry, and the sports drama film "Sugar", directed by Anna Boden and Ryan Fleck. She then co-starred in the 2011 romantic comedy film "Crazy, Stupid, Love" as Jordyn. From 2011 to 2012, she appeared on the seventh and eighth seasons of the Fox medical drama series "House", playing the role of Dominika Petrova, an immigrant who marries House in order to obtain a green card. Wydra then had a starring role in the 2012 fantasy thriller "After", alongside Steven Strait. The following year, she had a lead role as Katya Petrovna in the science fiction film "Europa Report". In March 2013, she was cast in the pilot of NBC's remake of the British drama series "Bad Girls".


title: Karolina Wydra

text: In 2013, Wydra joined the cast of HBO's fantasy drama series "True Blood" as Violet Mazurski, a vampire who becomes Jason Stackhouse's girlfriend. In the sixth season, Wydra was a recurring cast member, but was upgraded to the main cast for its seventh and final season. In 2014, Wydra recurred as the character Mara Paxton on the fifth season of FX's drama series "Justified", and guest starred as Agent Simone Taylor in an episode of the CBS drama series "Scorpion". Wydra then co-starred alongside Aaron Eckhart in the Blumhouse Productions horror film "Incarnate" (2015), directed by Brad Peyton. In 2015, she joined the cast of ABC's crime drama series "Wicked City" as Dianne Kubek, a beautiful detective working undercover as a barmaid and drug dealer on the Sunset Strip.


title: Karolina Wydra

text: In April 2016, it was announced that Wydra had been cast in the Showtime series revival of "Twin Peaks". In August 2016, she joined the Amazon Studios drama series "Sneaky Pete" in a recurring role.


title: Juliet Aubrey

text: Juliet Emma Aubrey (born 17 December 1966) is a British actress of theatre, film, and television. She won the 1995 BAFTA TV Award for Best Actress for playing Dorothea in the BBC serial "Middlemarch" (1994). She is also known for her role as Helen Cutter in the ITV series "Primeval" (2007–2011). Her film appearances include "Still Crazy" (1998), "The Constant Gardener" (2005) and "The Infiltrator" (2016).


title: Juliet Aubrey

text: The youngest of three siblings, Aubrey was born and brought up in Fleet, Hampshire. Aubrey attended King's College London from 1984, where she studied Classics and Archaeology. While there, however, her love of acting grew, and during a year studying in Italy where she joined a travelling theatre company, Aubrey decided to apply for drama school on her return. She went on to train for three years at the Central School of Speech and Drama.


title: Juliet Aubrey

text: Her first job was with the Oxford Stage Company playing Miranda in "The Tempest". Italian director Roberto Faenza gave Aubrey her first film role playing opposite Jean-Hugues Anglade in "Look to the Sky", a film produced by Elda Ferri, and set during the Nazi Holocaust. Antony Page and Louis Marks then cast Aubrey as Dorothea in the BBC adaptation of "Middlemarch" opposite Rufus Sewell, for which she won a BAFTA award for Best Actress, and the Broadcasting Press Guild for Best Actress. She then joined Haris Pasovic's Sarajevo Theatre Company. She appeared in several plays with the company, all created through the actors' improvisation. She continued to build her career as a theatre actress, appearing next in Trevor Nunn's "Summerfolk" and Katie Mitchell's "Ivanov" at the National Theatre, Tim Crouch's "An Oak Tree" for Karl James at the Soho Theatre, and "Three Sisters", "Twelfth Night" and "The Collection", all for Chris White. Michael Winterbottom then cast her opposite Robert Carlyle and James Nesbitt in the television film "Go Now".


title: Juliet Aubrey

text: Aubrey's subsequent films include Winterbottom's "Welcome to Sarajevo", Stephen Poliakoff's "Food of Love" — for which she won Best Actress at La Baule European Film Festival — Faenza's "Lost Lover", Giacomo Campiotti's "Time to Love", Richard Eyre's "Iris", Fernando Meirelles's "Constant Gardener" and Brian Gibson's "Still Crazy", nominated for two Golden Globes. Other features include Matt Lipsey's "Caught in the Act", and Mat Cod's "Super Eruption". Television work includes "The Village"; "The White Queen", "Criminal Justice", "Vera", "Hunted", and "Five Daughters". Her recent feature films are Scott Hicks's "Fallen"; Mitch Davis's "Stuck"; Fabio Guaglione's "Mine"; and Brad Furman's "Infiltrator". Aubrey played Lily Hill in the 2017 web television series "Snatch".


title: Juliet Aubrey

text: In 2001, Aubrey married production designer Steve Ritchie, whom she had met several years earlier while filming in Newcastle upon Tyne. They have two daughters.


title: Needy Streamer Overload

text: Needy Streamer Overload is a 2022 denpa-themed adventure visual novel created by Japanese developer Xemono and published by WSS Playground for macOS, Microsoft Windows, and Nintendo Switch. The player takes on the role of a manager for a female livestreamer, making decisions for her so that she can achieve her goal of reaching one million followers within a month. The game was initially titled Needy Girl Overdose, but this was changed in November 2021 in preparation for the western release. The game kept its original title in Japan.


title: Needy Streamer Overload

text: is a mentally ill young girl with a needy personality, who has dropped out of school and confined herself at home, living together with the self-insert player protagonist. In order to pay for rent, and to meet her parasocial attention needs, Ame decides to commence livestreaming on the internet, where she takes on the persona of , or for short, interacting with her stream viewers as she dons her wig and makeup. The protagonist, affectionately called , is tasked with managing her day-to-day life as she increases her follower count.


title: Needy Streamer Overload

text: The player interacts with Ame exclusively through a pastel-themed Windows 95-esque user interface, selecting her daily tasks via desktop icons, monitoring her stats via the Task Manager, and conversing with her through an instant messenger service called JINE. Each day is divided into noon, dusk and evening timeslots, and various actions can take up one or multiple of these timeslots. Ame can only livestream during the evening, as that is the time of day where stream viewers are most active. During Ame's livestreams, the player takes on the role of a content moderator responsible for deleting or promoting viewer comments within the stream's livechat. After streaming or concluding various activities, the player is able to obtain a glimpse of Ame's thoughts and mood via her posts on Tweeter.


title: Needy Streamer Overload

text: Alongside follower count, Ame has three stats which the player will need to monitor, namely stress, affection (towards the player), and mental darkness. Should certain attributes become too high or low, Ame will begin to display adverse effects. Activities that the player can choose for Ame may involve searching for new stream ideas, spending time together, sleeping, or abusing prescription and illicit drugs. Each of these activities affects Ame's stats in differing ways.


title: Needy Streamer Overload

text: At release, there were originally 22 different endings to the game, which are encountered based on the player's choices. Additional endings were added in an October 28, 2022 update coinciding with the game's Nintendo Switch release.


title: Needy Streamer Overload

text: Game development began in June 2020 under the initial title "Needy Girl Overdose". The title of the game for its English-language and Chinese-language releases were later changed to "Needy Streamer Overload" and "主播女孩重度依賴" () respectively; the reason for the title change was not disclosed. During the development process, the game was prototyped in Figma, and then built using the Unity engine. The game was initially scheduled for release during Spring 2021; eventually a release date of June 5, 2021 was announced, however was later delayed by the developers, citing the need for additional quality improvements. This seven month delay was later explained to have allowed the further implementation of additional game event scenarios, and a three- to four-fold increase in the number of animations. The game's dialogue has a wordcount of over 140,000 in the original Japanese script.


title: Needy Streamer Overload

text: , who had previously written various literature focused on mental illness, was responsible for the planning and writing of the game, while illustrator was responsible for the character designs. The game's art direction heavily draws influence from vaporwave aesthetics and retro pixel art from the PC-98 era, along with 1990s-era bishōjo games. There were originally four different female characters with unique personality traits planned during the early stages of development, however it was eventually decided that the final game would only feature a single heroine, combining various personality traits into the one character.


title: Needy Streamer Overload

text: Following the commercial success of the game's initial download-only Windows release, a Nintendo Switch port was announced in May 2022, including a Japan-exclusive physical release fanart book and a CD of the game's soundtrack. The Switch version was released on October 27, 2022, alongside a major update including new endings and text across platforms.


title: Needy Streamer Overload

text: The game's theme song is "Internet Overdose" by producer and vocalist Kotoko, composed and produced in a denpa song style. The song is available as a playable track in the music rhythm games "Muse Dash" and "Arcaea", as part of their respective collaboration updates. The theme has also charted on the Spotify "Japan Viral 50" chart.


title: Needy Streamer Overload

text: On April 4, 2022, publisher WSS Playground announced that in addition to being sold via online music distribution services, the game's soundtrack will also be released on 12-inch vinyl record format, to be released on June 29, 2022.


title: Needy Streamer Overload

text: A manga anthology titled began serialisation on ComicWalker and from December 23, 2022 onwards, featuring illustrations from Ui Shigure, Ohisashiburi, and 11 other artists.


title: Needy Streamer Overload

text: "Needy Streamer Overload" sold over 100,000 copies during its first week of release. As of June 2022, the game has sold over 500,000 copies on Steam. The Nintendo Switch version sold 11,693 copies within its first week of release in Japan, making it the tenth bestselling retail game of the week in the country. It has a rating of "overwhelmingly positive" based on more than 5,800 user reviews.


title: Needy Streamer Overload

text: "IGN Japan" notes that the game has merit as a work of satire intended at picking apart modern livestreaming culture and toxic relationships, and praises the game's art and presentation, however criticises the limited choices and gimmicks available to express the player character's persona in different ways, arguing that the player is not given the opportunity to become properly aware of any consequences from their interactions with Ame. In addition, the character of Ame is criticised as more of a "male nerd's delusion" of what girls are like, rather than how women realistically behave.


title: Needy Streamer Overload

text: "Dengeki Online" suggests that despite the gratuitous use of mentally disturbing content, "Needy Streamer Overload" is a quality piece of work full of expressiveness that can only be delivered through games as an entertainment medium, while also pointing out that players unfamiliar with internet culture may not fully appreciate what the game attempts to portray.


title: Needy Streamer Overload

text: A review from "" mentions that while the game has a few rough areas and bugs which detract from the experience, the game still represents a domestic Japanese indie title with a high degree of preciseness put into it, closely satirising the intended audience's internet addiction while at the same time acknowledging how it forms part of their identity.


title: Needy Streamer Overload

text: Another review published by "United Daily News" in Taiwan compares the game progression experience of "Needy Streamer Overload" to that of "Undertale", in reference to how the player learns more and more about Ame during each subsequent playthrough to unlock all the endings. While acknowledging that that game's themes may be too heavy for some audiences, it claims that the game is a "masterpiece" for players who enjoy the premise of exploring "menhera" issues and are fans of "denpa" culture, and also praised the official Chinese language localisation of the game, especially in regards to its use of Chinese internet memes, claiming that they accurately express the mood of the original Japanese script while still maintaining some degree of familiarity among the audience.


title: Needy Streamer Overload

text: " provides a comparatively more critical review of the game, claiming that it is a disturbing and morally contentious title which uses ambiguity to hide the player's evident hatred towards Ame, as observable through the player's ability to torment her, and how the game rewards players who choose to ignore repercussions on Ame's mental health when allowing her to overdose on drugs and slice her wrists in order to progress through the game and unlock new scenarios to explore. The review also criticises what it views as the game's lack of sagacity when utilising shock value, suggesting that such usage moves beyond parodic intent.


title: Needy Streamer Overload

text: South Korean gaming magazine " praised the game's chiptune soundtrack and overall atmosphere, however criticises the game's short length and the repetitive nature of progressing multiple playthroughs to unlock each ending, claiming that unlike other "raising simulators" such as "Princess Maker" which feel rewarding as the game goes on, "Needy Streamer Overload" may not provide the player with as much of a sense of accomplishment, despite being a game with a similar premise.


title: Needy Streamer Overload

text: "Needy Streamer Overload" won the "Most Stream-friendly Game" and "Best New Characters" awards at the 2022 in Japan. The game also was one of two runner-up titles receiving honourable mention for the Game Designers Award at the 2022 Japan Game Awards.


title: Ramsey's theorem

text: In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .)


title: Ramsey's theorem

text: Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of "monochromatic subsets", that is, subsets of connected edges of just one colour.


title: Ramsey's theorem

text: An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours, , and any given integers , there is a number, , such that if the edges of a complete graph of order are coloured with different colours, then for some between 1 and , it must contain a complete subgraph of order whose edges are all colour . The special case above has (and and ).


title: Ramsey's theorem

text: Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex, . There are 5 edges incident to and so (by the pigeonhole principle) at least 3 of them must be the same colour. Without loss of generality we can assume at least 3 of these edges, connecting the vertex, , to vertices, , and , are blue. (If not, exchange red and blue in what follows.) If any of the edges, , , , are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring, "any" contains a monochromatic , and therefore . The popular version of this is called the theorem on friends and strangers.


title: Ramsey's theorem

text: An alternative proof works by double counting. It goes as follows: Count the number of ordered triples of vertices, , , , such that the edge, , is red and the edge, , is blue. Firstly, any given vertex will be the middle of either (all edges from the vertex are the same colour), (four are the same colour, one is the other colour), or (three are the same colour, two are the other colour) such triples. Therefore, there are at most such triples. Secondly, for any non-monochromatic triangle , there exist precisely two such triples. Therefore, there are at most 18 non-monochromatic triangles. Therefore, at least 2 of the 20 triangles in the are monochromatic.


title: Ramsey's theorem

text: Conversely, it is possible to 2-colour a without creating any monochromatic , showing that . The unique colouring is shown to the right. Thus .


title: Ramsey's theorem

text: The task of proving that was one of the problems of William Lowell Putnam Mathematical Competition in 1953, as well as in the Hungarian Math Olympiad in 1947.


title: Ramsey's theorem

text: A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely and .


title: Ramsey's theorem

text: Suppose that we have an edge colouring of a complete graph using 3 colours, red, green and blue. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex . Consider the set of vertices that have a red edge to the vertex . This is called the red neighbourhood of . The red neighbourhood of cannot contain any red edges, since otherwise there would be a red triangle consisting of the two endpoints of that red edge and the vertex . Thus, the induced edge colouring on the red neighbourhood of has edges coloured with only two colours, namely green and blue. Since , the red neighbourhood of can contain at most 5 vertices. Similarly, the green and blue neighbourhoods of can contain at most 5 vertices each. Since every vertex, except for itself, is in one of the red, green or blue neighbourhoods of , the entire complete graph can have at most vertices. Thus, we have .


title: Ramsey's theorem

text: To see that , it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on , the so-called untwisted and twisted colourings. Both colourings are shown in the figures to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom.


title: Ramsey's theorem

text: If we select any colour of either the untwisted or twisted colouring on , and consider the graph whose edges are precisely those edges that have the specified colour, we will get the Clebsch graph.


title: Ramsey's theorem

text: It is known that there are exactly two edge colourings with 3 colours on that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on , respectively.


title: Ramsey's theorem

text: It is also known that there are exactly 115 edge colourings with 3 colours on that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.


title: Ramsey's theorem

text: The theorem for the 2-colour case can be proved by induction on . It is clear from the definition that for all , . This starts the induction. We prove that exists by finding an explicit bound for it. By the inductive hypothesis and exist.


title: Ramsey's theorem

text: "Proof." Consider a complete graph on vertices whose edges are coloured with two colours. Pick a vertex from the graph, and partition the remaining vertices into two sets and , such that for every vertex , is in if edge is blue, and is in if is red. Because the graph has formula_2 vertices, it follows that either formula_3 or formula_4 In the former case, if has a red then so does the original graph and we are finished. Otherwise has a blue and so formula_5 has a blue by the definition of . The latter case is analogous. Thus the claim is true and we have completed the proof for 2 colours.


title: Ramsey's theorem

text: "Proof". Suppose and are both even. Let and consider a two-coloured graph of vertices. If is degree of -th vertex in the blue subgraph, then, according to the Handshaking lemma, formula_7 is even. Given that is odd, there must be an even . Assume is even, and are the vertices incident to vertex 1 in the blue and red subgraphs, respectively. Then both formula_8 and formula_9 are even. According to the Pigeonhole principle, either formula_10 or formula_11 Since is even, while is odd, the first inequality can be strengthened, so either formula_12 or formula_11 Suppose formula_14 Then either the subgraph has a red and the proof is complete, or it has a blue which along with vertex 1 makes a blue . The case formula_15 is treated similarly.


title: Ramsey's theorem

text: "Proof." Consider a complete graph of formula_17 vertices and colour its edges with colours. Now 'go colour-blind' and pretend that and are the same colour. Thus the graph is now -coloured. Due to the definition of formula_18 such a graph contains either a mono-chromatically coloured with colour for some or a -coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of we must have either a -monochrome or a -monochrome . In either case the proof is complete.


title: Ramsey's theorem

text: Lemma 1 implies that any is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for colours in terms of Ramsey numbers for fewer colours. Therefore any is finite for any number of colours. This proves the theorem.


title: Ramsey's theorem

text: The numbers in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. The Ramsey number, , gives the solution to the party problem, which asks the minimum number of guests, , that must be invited so that at least will know each other or at least will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices, , such that all undirected simple graphs of order , contain a clique of order , or an independent set of order . Ramsey's theorem states that such a number exists for all and .


title: Ramsey's theorem

text: By symmetry, it is true that . An upper bound for can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. There are also very few numbers and for which we know the exact value of .


title: Ramsey's theorem

text: Computing a lower bound for usually requires exhibiting a blue/red colouring of the graph with no blue subgraph and no red subgraph. Such a counterexample is called a "Ramsey graph". Brendan McKay maintains a list of known Ramsey graphs. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence.


title: Ramsey's theorem

text: A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. Each complete graph has edges, so there would be a total of graphs to search through (for colours) if brute force is used. Therefore, the complexity for searching all possible graphs (via brute force) is for colourings and at most nodes.


title: Ramsey's theorem

text: The situation is unlikely to improve with the advent of quantum computers. The best known algorithm exhibits only a quadratic speedup (c.f. Grover's algorithm) relative to classical computers, so that the computation time is still exponential in the number of nodes.


title: Ramsey's theorem

text: As described above, . It is easy to prove that , and, more generally, that for all : a graph on nodes with all edges coloured red serves as a counterexample and proves that ; among colourings of a graph on nodes, the colouring with all edges coloured red contains a -node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.)


title: Ramsey's theorem

text: Using induction inequalities, it can be concluded that , and therefore . There are only two graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among different 2-colourings of 16-node graphs, and only one graph (the Paley graph of order 17) among colourings. (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that .


title: Ramsey's theorem

text: The exact value of is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)) and 48 (Angeltveit and McKay (2017)) (inclusive).


title: Ramsey's theorem

text: In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that . They were able to construct exactly 656 graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a graph.


title: Ramsey's theorem

text: For with , only weak bounds are available. Lower bounds for and have not been improved since 1965 and 1972, respectively.


title: Ramsey's theorem

text: The standard survey on the development of Ramsey number research is the "Dynamic Survey 1" of the "Electronic Journal of Combinatorics", by Stanisław Radziszowski, which is periodically updated. Where not cited otherwise, entries in the table below are taken from the January 2021 edition. (Note there is a trivial symmetry across the diagonal since .)


title: Ramsey's theorem

text: was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for , this gives . Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at and respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at


title: Ramsey's theorem

text: For the off-diagonal Ramsey numbers , it is known that they are of order ; this may be stated equivalently as saying that the smallest possible independence number in an -vertex triangle-free graph is


title: Ramsey's theorem

text: The upper bound for is given by Ajtai, Komlós, and Szemerédi, the lower bound was obtained originally by Kim, and was improved by Griffiths, Morris, Fiz Pontiveros, and Bohman and Keevash, by analysing the triangle-free process. More generally, for off-diagonal Ramsey numbers, , with fixed and growing, the best known bounds are


title: Ramsey's theorem

text: There is a less well-known yet interesting analogue of Ramsey's theorem for induced subgraphs. Roughly speaking, instead of finding a monochromatic subgraph, we are now required to find a monochromatic induced subgraph. In this variant, it is no longer sufficient to restrict our focus to complete graphs, since the existence of a complete subgraph does not imply the existence of an induced subgraph. The qualitative statement of the theorem in the next section was first proven independently by Erdős, Hajnal and Pósa, Deuber and Rödl in the 1970s. Since then, there has been much research in obtaining good bounds for induced Ramsey numbers.


title: Ramsey's theorem

text: Let be a graph on vertices. Then, there exists a graph such that any coloring of the edges of using two colors contains a monochromatic induced copy of (i.e. an induced subgraph of such that it is isomorphic to and its edges are monochromatic). The smallest possible number of vertices of is the induced Ramsey number .


title: Ramsey's theorem

text: Sometimes, we also consider the asymmetric version of the problem. We define to be the smallest possible number of vertices of a graph such that every coloring of the edges of using only red or blue contains a red induced subgraph of or blue induced subgraph of .


title: Ramsey's theorem

text: Similar to Ramsey's Theorem, it is unclear a priori whether induced Ramsey numbers exist for every graph . In the early 1970s, Erdős, Hajnal and Pósa, Deuber and Rödl independently proved that this is the case. However, the original proofs gave terrible bounds (e.g. towers of twos) on the induced Ramsey numbers. It is interesting to ask if better bounds can be achieved. In 1974, Paul Erdős conjectured that there exists a constant such that every graph on vertices satisfies . If this conjecture is true, it would be optimal up to the constant because the complete graph achieves a lower bound of this form (in fact, it's the same as Ramsey numbers). However, this conjecture is still open as of now.


title: Ramsey's theorem

text: However, that was still far from the exponential bound conjectured by Erdős. It was not until 1998 when a major breakthrough was achieved by Kohayakawa, Prömel and Rödl, who proved the first almost-exponential bound of for some constant . Their approach was to consider a suitable random graph constructed on projective planes and show that it has the desired properties with nonzero probability. The idea of using random graphs on projective planes have also previously been used in studying Ramsey properties with respect to vertex colorings and the induced Ramsey problem on bounded degree graphs .


title: Ramsey's theorem

text: Kohayakawa, Prömel and Rödl's bound remained the best general bound for a decade. In 2008, Fox and Sudakov provided an explicit construction for induced Ramsey numbers with the same bound. In fact, they showed that every -graph with small and suitable contains an induced monochromatic copy of any graph on vertices in any coloring of edges of in two colors. In particular, for some constant , the Paley graph on vertices is such that all of its edge colorings in two colors contain an induced monochromatic copy of every -vertex graph.


title: Ramsey's theorem

text: In 2010, Conlon, Fox and Sudakov were able to improve the bound to , which remains the current best upper bound for general induced Ramsey numbers. Similar to the previous work in 2008, they showed that every -graph with small and edge density contains an induced monochromatic copy of every graph on vertices in any edge coloring in two colors. Currently, Erdős's conjecture that remains open and is one of the important problems in extremal graph theory.


title: Ramsey's theorem

text: For lower bounds, not much is known in general except for the fact that induced Ramsey numbers must be at least the corresponding Ramsey numbers. Some lower bounds have been obtained for some special cases (see Special Cases).


title: Ramsey's theorem

text: While the general bounds for the induced Ramsey numbers are exponential in the size of the graph, the behaviour is much different on special classes of graphs (in particular, sparse ones). Many of these classes have induced Ramsey numbers polynomial in the number of vertices.


title: Ramsey's theorem

text: If is a tree on vertices, it is known that . It is also known that is superlinear (i.e. ). Note that this is in contrast to the usual Ramsey numbers, where the Burr–Erdős conjecture (now proven) tells us that is linear (since trees are 1-degenerate).


title: Ramsey's theorem

text: For graphs with number of vertices and bounded degree , it was conjectured that , for some constant depending only on . This result was first proven by Łuczak and Rödl in 1996, with growing as a tower of twos with height . More reasonable bounds for were obtained since then. In 2013, Conlon, Fox and Zhao showed using a counting lemma for sparse pseudorandom graphs that , where the exponent is best possible up to constant factors.


title: Ramsey's theorem

text: Similar to Ramsey numbers, we can generalize the notion of induced Ramsey numbers to hypergraphs and multicolor settings.


title: Ramsey's theorem

text: We can also generalize the induced Ramsey's theorem to a multicolor setting. For graphs , define to be the minimum number of vertices in a graph such that any coloring of the edges of into colors contain an induced subgraph isomorphic to where all edges are colored in the -th color for some . Let ( copies of ).


title: Ramsey's theorem

text: It is possible to derive a bound on which is approximately a tower of two of height by iteratively applying the bound on the two-color case. The current best known bound is due to Fox and Sudakov, which achieves , where is the number of vertices of and is a constant depending only on .


title: Ramsey's theorem

text: We can extend the definition of induced Ramsey numbers to -uniform hypergraphs by simply changing the word "graph" in the statement to "hypergraph". Furthermore, we can define the multicolor version of induced Ramsey numbers in the same way as the previous subsection.


title: Ramsey's theorem

text: Let be a -uniform hypergraph with vertices. Define the tower function by letting and for , . Using the hypergraph container method, Conlon, Dellamonica, La Fleur, Rödl and Schacht were able to show that for , for some constant depending on only and . In particular, this result mirrors the best known bound for the usual Ramsey number when .


title: Ramsey's theorem

text: A further result, also commonly called "Ramsey's theorem", applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in set-theoretic terminology.


title: Ramsey's theorem

text: "Proof": The proof is by induction on , the size of the subsets. For , the statement is equivalent to saying that if you split an infinite set into a finite number of sets, then one of them is infinite. This is evident. Assuming the theorem is true for , we prove it for . Given a -colouring of the -element subsets of , let be an element of and let We then induce a -colouring of the -element subsets of , by just adding to each -element subset (to get an -element subset of ). By the induction hypothesis, there exists an infinite subset of such that every -element subset of is coloured the same colour in the induced colouring. Thus there is an element and an infinite subset such that all the -element subsets of consisting of and elements of have the same colour. By the same argument, there is an element in and an infinite subset of with the same properties. Inductively, we obtain a sequence such that the colour of each -element subset with depends only on the value of . Further, there are infinitely many values of such that this colour will be the same. Take these 's to get the desired monochromatic set.


title: Ramsey's theorem

text: A stronger but unbalanced infinite form of Ramsey's theorem for graphs, the Erdős–Dushnik–Miller theorem, states that every infinite graph contains either a countably infinite independent set, or an infinite clique of the same cardinality as the original graph.


title: Ramsey's theorem

text: It is possible to deduce the finite Ramsey theorem from the infinite version by a proof by contradiction. Suppose the finite Ramsey theorem is false. Then there exist integers , , such that for every integer , there exists a -colouring of without a monochromatic set of size . Let denote the -colourings of without a monochromatic set of size .


title: Ramsey's theorem

text: For any , the restriction of a colouring in to (by ignoring the colour of all sets containing ) is a colouring in . Define to be the colourings in which are restrictions of colourings in . Since is not empty, neither is .


title: Ramsey's theorem

text: Similarly, the restriction of any colouring in is in , allowing one to define as the set of all such restrictions, a non-empty set. Continuing so, define for all integers , .


title: Ramsey's theorem

text: Then every colouring in is the restriction of a colouring in . Therefore, by unrestricting a colouring in to a colouring in , and continuing doing so, one constructs a colouring of formula_29 without any monochromatic set of size . This contradicts the infinite Ramsey theorem.


title: Ramsey's theorem

text: If a suitable topological viewpoint is taken, this argument becomes a standard compactness argument showing that the infinite version of the theorem implies the finite version.


title: Ramsey's theorem

text: The theorem can also be extended to hypergraphs. An -hypergraph is a graph whose "edges" are sets of vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers and , and any integers , there is an integer such that if the hyperedges of a complete -hypergraph of order are coloured with different colours, then for some between 1 and , the hypergraph must contain a complete sub--hypergraph of order whose hyperedges are all colour . This theorem is usually proved by induction on , the 'hyper-ness' of the graph. The base case for the proof is , which is exactly the theorem above.


title: Ramsey's theorem

text: For we know the exact value of one non-trivial Ramsey number, namely . This fact was established by Brendan McKay and Stanisław Radziszowski in 1991. Additionally, we have: , and .


title: Ramsey's theorem

text: It is also possible to define Ramsey numbers for "directed" graphs; these were introduced by . Let be the smallest number such that any complete graph with singly directed arcs (also called a "tournament") and with nodes contains an acyclic (also called "transitive") -node subtournament.


title: Ramsey's theorem

text: This is the directed-graph analogue of what (above) has been called , the smallest number such that any 2-colouring of the edges of a complete "un"directed graph with nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc "colours" is the two "directions" of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way"; i.e., "acyclic.")


title: Ramsey's theorem

text: In terms of the partition calculus Ramsey's theorem can be stated as formula_30 for all finite "n" and "k". A Ramsey cardinal, formula_31, is a large cardinal axiomatically defined to satisfy the related formula: formula_32.


title: Ramsey's theorem

text: In reverse mathematics, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs (the case "n" = 2) and for infinite multigraphs (the case "n" ≥ 3). The multigraph version of the theorem is equivalent in strength to the arithmetical comprehension axiom, making it part of the subsystem ACA of second-order arithmetic, one of the big five subsystems in reverse mathematics. In contrast, by a theorem of David Seetapun, the graph version of the theorem is weaker than ACA, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems. Over ZF, however, the graph version is equivalent to the classical Kőnig's lemma.