Ramsey's theorem

Ramsey's theorem
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 .)
Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by Frank 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.
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 ).
Examples
R(3, 3) = 6
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.
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.
Conversely, it is possible to 2-colour a without creating any monochromatic , showing that . The unique colouring is shown to the right. Thus .
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.
A multicolour example: R(3, 3, 3) = 17
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 .
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 .
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 left, and the twisted colouring on the right.
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.
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.
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.
Proof
2-colour case
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.
Lemma 1.
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 vertices, it follows that either or In the former case, if has a red then so does the original graph and we are finished. Otherwise has a blue and so 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.
In this 2-colour case, if and are both even, the induction inequality can be strengthened to:
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, 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 and are even. According to the Pigeonhole principle, either or Since is even, while is odd, the first inequality can be strengthened, so either or Suppose 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 is treated similarly.
Case of more colours
Lemma 2. If , then
Proof. Consider a complete graph of 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 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.
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.
Ramsey numbers
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 .
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 .
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.
Computational complexity
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.
The situation is unlikely to improve with the advent of quantum computers. One of the best-known searching algorithms for unstructured datasets 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.
Known values
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.)
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 .
The fact that was first established by Brendan McKay and Stanisław Radziszowski in 1995.
The exact value of is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)) and 48 (Angeltveit and McKay (2017)) (inclusive).
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.
For with , only weak bounds are available. Lower bounds for and have not been improved since 1965 and 1972, respectively.
with are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. with are given by and for all values of .
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 .)
Asymptotics
The inequality may be applied inductively to prove that
In particular, this result, due to Erdős and Szekeres, implies that when ,
An exponential lower bound,
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, the exponential growth factors of either bound were not improved for a long time, and that for the lower bound still stands at . There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers are
due to Spencer and Conlon respectively, while a 2023 preprint by Morris, Campos, Griffiths and Sahasrabudhe claims to have made exponential progress using an algorithmic construction relaying on a graph structure dubbed books, improving the upper bound to
with and where it is believed these parameters could be optimized in particular .
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
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, and more broadly these works set the best known asymptotic bounds on for general off-diagonal Ramsey numbers,
For the bounds become , but a 2023 preprint has improved the lower bound to which settles a question of Erdos who offered 250 dollars for a proof that the lower limit has form .
Induced Ramsey
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.
Statement
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 .
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 .
History and bounds
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.
In 1984, Erdős and Hajnal claimed that they proved the bound
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 .
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.
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.
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).
Special cases
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.
If is a cycle, path or star on vertices, it is known that is linear in .
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).
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.
Generalizations
Similar to Ramsey numbers, we can generalize the notion of induced Ramsey numbers to hypergraphs and multicolor settings.
More colors
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 ).
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 .
Hypergraphs
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.
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 .
Extensions of the theorem
Infinite graphs
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.
Theorem. Let be some infinite set and colour the elements of (the subsets of of size ) in different colours. Then there exists some infinite subset of such that the size subsets of all have the same colour.
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.
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.
Infinite version implies the finite
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 .
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 .
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 , .
Now, for any integer ,
and each set is non-empty. Furthermore, is finite as
It follows that the intersection of all of these sets is non-empty, and let
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 without any monochromatic set of size . This contradicts the infinite Ramsey theorem.
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.
Hypergraphs
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.
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 .
Directed graphs
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.
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 undirected 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.")
We have , and .
Ramsey cardinals
In terms of the partition calculus Ramsey's theorem can be stated as for all finite n and k. A Ramsey cardinal is a large cardinal axiomatically defined to satisfy the related formula: .
Relationship to the axiom of choice
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 ACA0 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 ACA0, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems. Over ZF, however, the graph version implies the classical Kőnig's lemma, whereas the converse implication does not hold, since Kőnig's lemma is equivalent to countable choice from finite sets in this setting.
See also
Ramsey cardinal
Paris–Harrington theorem
Sim (pencil game)
Infinite Ramsey theory
Van der Waerden number
Ramsey game
Erdős–Rado theorem
Notes
References
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External links
Ramsey@Home is a distributed computing project designed to find new lower bounds for various Ramsey numbers using a host of different techniques.
The Electronic Journal of Combinatorics dynamic survey of small Ramsey numbers (by Stanisław Radziszowski)
Ramsey Number – from MathWorld (contains lower and upper bounds up to R(19, 19))
Ramsey Number – Geoffrey Exoo (Contains R(5, 5) > 42 counter-proof)
Ramsey theory
Theorems in graph theory
Articles containing proofs