In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges  an -vertex graph can have such that it does not have a subgraph isomorphic to . In this context,  is called a forbidden subgraph.

An equivalent problem is how many edges in an -vertex graph guarantee that it has a subgraph isomorphic to ?

Definitions 
The extremal number  is the maximum number of edges in an -vertex graph containing no subgraph isomorphic to .  is the complete graph on  vertices.  is the Turán graph: a complete -partite graph on  vertices, with vertices distributed between parts as equally as possible. The chromatic number  of  is the minimum number of colors needed to color the vertices of  such that no two adjacent vertices have the same color.

Upper bounds

Turán's theorem 

Turán's theorem states that for positive integers  satisfying , 

This solves the forbidden subgraph problem for . Equality cases for Turán's theorem come from the Turán graph .

This result can be generalized to arbitrary graphs  by considering the chromatic number  of . Note that  can be colored with  colors and thus has no subgraphs with chromatic number greater than . In particular,  has no subgraphs isomorphic to . This suggests that the general equality cases for the forbidden subgraph problem may be related to the equality cases for . This intuition turns out to be correct, up to  error.

Erdős–Stone theorem 

Erdős–Stone theorem states that for all positive integers  and all graphs , 

When  is not bipartite, this gives us a first-order approximation of .

Bipartite graphs 
For bipartite graphs , the Erdős–Stone theorem only tells us that . The forbidden subgraph problem for bipartite graphs is known as the Zarankiewicz problem, and it is unsolved in general.

Progress on the Zarankiewicz problem includes following theorem:

Kővári–Sós–Turán theorem. For every pair of positive integers  with , there exists some constant  (independent of ) such that  for every positive integer .

Another result for bipartite graphs is the case of even cycles, . Even cycles are handled by considering a root vertex and paths branching out from this vertex. If two paths of the same length  have the same endpoint and do not overlap, then they create a cycle of length . This gives the following theorem.

Theorem (Bondy and Simonovits, 1974). There exists some constant  such that  for every positive integer  and positive integer .

A powerful lemma in extremal graph theory is dependent random choice. This lemma allows us to handle bipartite graphs with bounded degree in one part:

Theorem (Alon, Krivelevich, and Sudakov, 2003). Let  be a bipartite graph with vertex parts  and  such that every vertex in  has degree at most . Then there exists a constant  (dependent only on ) such that for every positive integer .

In general, we have the following conjecture.:

Rational Exponents Conjecture (Erdős and Simonovits). For any finite family  of graphs, if there is a bipartite , then there exists a rational  such that .

A survey by Füredi and Simonovits describes progress on the forbidden subgraph problem in more detail.

Lower bounds 
There are various techniques used for obtaining the lower bounds.

Probabilistic method 
While this method mostly gives weak bounds, the theory of random graphs is a rapidly developing subject. It is based on the idea that if we take a graph randomly with a sufficiently small density, the graph would contain only a small number of subgraphs of  inside it. These copies can be removed by removing one edge from every copy of  in the graph, giving us a  free graph.  

The probabilistic method can be used to prove where  is a constant only depending on the graph . For the construction we can take the Erdős-Rényi random graph , that is the graph with  vertices and the edge been any two vertices drawn with probability , independently. After computing the expected number of copies of  in  by linearity of expectation, we remove one edge from each such copy of  and we are left with a -free graph in the end. The expected number of edges remaining can be found to be  for a constant  depending on . Therefore, at least one -vertex graph exists with at least as many edges as the expected number.

This method can also be used to find the constructions of a graph for bounds on the girth of the graph. The girth, denoted by , is the length of the shortest cycle of the graph. Note that for , the graph must forbid all the cycles with length less than equal to . By linearity of expectation,the expected number of such forbidden cycles is equal to the sum of the expected number of cycles  (for .). We again remove the edges from each copy of a forbidden graph and end up with a graph free of smaller cycles and , giving us  edges in the graph left.

Algebraic constructions 
For specific cases, improvements have been made by finding algebraic constructions. A common feature for such constructions is that it involves the use of geometry to construct a graph, with vertices representing geometric objects and edges according to the algebraic relations between the vertices. We end up with no subgraph of , purely due to purely geometric reasons, while the graph has a large number of edges to be a strong bound due to way the incidences were defined. The following proof by Erdős, Rényi, and Sős establishing the lower bound on  as, demonstrates the power of this method.

First, suppose that  for some prime . Consider the polarity graph  with vertices elements of  and edges between vertices  and  if and only if  in . This graph is -free because a system of two linear equations in  cannot have more than one solution. A vertex  (assume ) is connected to  for any , for a total of at least  edges (subtracted 1 in case ). So there are at least  edges, as desired. For general , we can take  with  (which is possible because there exists a prime  in the interval for sufficiently large ) and construct a polarity graph using such , then adding  isolated vertices, which do not affect the asymptotic value. 

The following theorem is a similar result for .

Theorem (Brown, 1966). 
Proof outline. Like in the previous theorem, we can take  for prime  and let the vertices of our graph be elements of . This time, vertices  and  are connected if and only if  in , for some specifically chosen . Then this is -free since at most two points lie in the intersection of three spheres. Then since the value of  is almost uniform across , each point should have around  edges, so the total number of edges is .

However, it remains an open question to tighten the lower bound for  for .

Theorem (Alon et al., 1999) For ,

Randomized Algebraic constructions 
This technique combines the above two ideas. It uses random polynomial type relations when defining the incidences between vertices, which are in some algebraic set. Using this technique to prove the following theorem.

Theorem: For every , there exists some  such that .

Proof outline: We take the largest prime power  with . Due to the prime gaps, we have .  Let  be a random polynomial in  with degree at most  in  and  and satisfying . Let the graph  have the vertex set  such that two vertices  are adjacent if . 

We fix a set , and defining a set  as the elements of  not in  satisfying  for all elements . By the Lang–Weil bound, we obtain that for  sufficiently large enough, we have  or  for some constant .Now, we compute the expected number of  such that  has size greater than , and  remove a vertex from each such . The resulting graph turns out to be  free, and at least one graph exists with the expectation of the number of edges of this resulting graph.

Supersaturation 

Supersaturation refers to a variant of the forbidden subgraph problem, where we consider when some -uniform graph  contains many copies of some forbidden subgraph . Intuitively, one would expect this to once  contains significantly more than  edges. We introduce Turán density to formalize this notion.

Turán density 

The Turán density of a -uniform graph  is defined to be 

 

It is true that  is in fact positive and monotone decreasing, so the limit must therefore exist. 

As an example, Turán's Theorem gives that , and the Erdős–Stone theorem gives that . In particular, for bipartite , . Determining the Turán density  is equivalent to determining  up to an  error.

Supersaturation Theorem 

Consider an -uniform hypergraph  with  vertices. The supersaturation theorem states that for every , there exists a  such that if  is a graph on  vertices and at least  edges for  sufficiently large, then there are at least  copies of . 

Equivalently, we can restate this theorem as the following: If a graph  with  vertices has  copies of , then there are at most  edges in .

Applications 

We may solve various forbidden subgraph problems by considering supersaturation-type problems. We restate and give a proof sketch of the Kővári–Sós–Turán theorem below: 

Kővári–Sós–Turán theorem. For every pair of positive integers  with , there exists some constant  (independent of ) such that  for every positive integer .
Proof. Let  be a -graph on  vertices, and consider the number of copies of  in . Given a vertex of degree , we get exactly  copies of  rooted at this vertex, for a total of  copies. Here,  when . By convexity, there are at total of at least  copies of . Moreover, there are clearly  subsets of  vertices, so if there are more than  copies of , then by the Pigeonhole Principle there must exist a subset of  vertices which form the set of leaves of at least  of these copies, forming a . Therefore, there exists an occurrence of  as long as we have . In other words, we have an occurrence if , which simplifies to , which is the statement of the theorem. 

In this proof, we are using the supersaturation method by considering the number of occurrences of a smaller subgraph. Typically, applications of the supersaturation method do not use the supersaturation theorem. Instead, the structure often involves finding a subgraph  of some forbidden subgraph  and showing that if it appears too many times in , then  must appear in  as well. Other theorems regarding the forbidden subgraph problem which can be solved with supersaturation include:

Generalizations 
The problem may be generalized for a set of forbidden subgraphs :  find the maximal number of edges in an -vertex graph which does not have a subgraph isomorphic to any graph from .

There are also hypergraph versions of forbidden subgraph problems that are much more difficult. For instance, Turán's problem may be generalized to asking for the largest number of edges in an -vertex 3-uniform hypergraph that contains no tetrahedra. The analog of the Turán construction would be to partition the vertices into almost equal subsets , and connect vertices  by a 3-edge if they are all in different s, or if two of them are in  and the third is in  (where ). This is tetrahedron-free, and the edge density is . However, the best known upper bound is 0.562, using the technique of flag algebras.

See also 
Biclique-free graph
Erdős–Hajnal conjecture
Turán number
Subgraph isomorphism problem
Forbidden graph characterization

References 

Extremal graph theory