Ramsey's Theorem

 Ramsey's Theorem

 Ramsey's Theorem is a fundamental result in graph theory that provides a bound on the size of graphs that can be guaranteed to have certain properties. Specifically, it states that for any positive integers k, r, there exists a smallest positive integer R(k,r) such that any graph with at least R(k,r) vertices contains either a k-clique (a complete subgraph of k vertices) or an independent set of size r (a subset of r vertices with no edges between them). In other words, any large enough graph must either contain a densely connected subgraph or a sparsely connected subgraph.

To prove Ramsey's Theorem, we will use mathematical induction on the number of vertices of the graph. We will show that if a graph G on n vertices does not contain a k-clique or an independent set of size r, then it is possible to add a new vertex to G and still not have either of these subgraphs.

Base Case: Let n = 1. A graph on one vertex does not contain a k-clique or an independent set of size r, since there are no edges, and hence R(k,r) = 1.

Inductive Step: Assume that for some integer n ≥ 1, the theorem holds for all graphs on n vertices. Let G be a graph on n+1 vertices that does not contain a k-clique or an independent set of size r. Consider any vertex v in G. Let G_v be the subgraph of G induced by the vertices other than v. By the induction hypothesis, G_v does not contain a k-clique or an independent set of size r. Thus, by the definition of R(k,r), we have:

R(k,r) > max{|G_v|, R(k-1, r), R(k, r-1)}

where |G_v| denotes the number of vertices in G_v.

Let s = max{|G_v|, R(k-1, r), R(k, r-1)}. We claim that R(k,r) ≤ R(k-1, r) + R(k, r-1) for all positive integers k,r. If we can prove this, then we have:

|R(k,r)| > s ≥ max{|G_v|, R(k-1, r), R(k, r-1)}

which implies that G has either a k-clique or an independent set of size r.

To prove the claim, consider any graph H on at least R(k-1, r) + R(k, r-1) vertices. By the definition of R(k-1, r) and R(k, r-1), we know that H either contains a k-1 clique or an independent set of size r, respectively. Let A be the vertices in H that form a k-1 clique, and let B be the vertices in H that form an independent set of size r. If A has at least R(k-1, r) vertices, then it contains a k-clique, since H is a complete graph on A. Otherwise, B has at least R(k, r-1) vertices, and it forms an independent set, since H has no edges between the vertices in B. Therefore, we have shown that any graph on at least R(k-1, r) + R(k, r-1) vertices contains either a k-clique or an independent set of size r. Thus, we have:

R(k,r) ≤ R(k-1, r) + R(k, r-1)

This completes the proof of Ramsey's Theorem using mathematical induction.