Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex, v. There are 5 edges … 展开

The theorem for the 2-colour case can be proved by induction on r + s. It is clear from the definition that for all n, R(n, 2) = R(2, n) = n. This starts the induction. We prove that R(r, s) exists by finding an explicit bound for it. By the inductiv… 展开

The numbers R(r, s) in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. The Ramsey number R(m, n) gives the solution to the party problem, which asks the minimum nu… 展开

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 subgrap… 展开

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 … 展开

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 versi… 展开