For example, suppose you have a complete graph of order n; this means you have n vertices (dots) where each vertex is connected to every other vertex by an edge (a line). A complete graph of order 3 is called a triangle. Color every edge red or blue. How large must n be in order to ensure that there is either a blue triangle or a red triangle; that is, in order to guarantee that there are three vertices each of which is connected to the others by edges of the same color? It turns out that the answer is 6. See the Ramsey's theorem article for a rigorous proof.
Another way to interpret this: at any party with at least six people, there are either three people who are all mutual acquaintances (each one knows the other two) or mutual strangers (each one does not know either of the other two).
This is a special case of Ramsey's theorem, which says that for any given integer c,any given integers n_{1},...,n_{c}, there is a number, R(n_{1},...,n_{c};c), such that if the edges of a complete graph of order R(n_{1},...,n_{c};c) are colored with c different colors, then for some i between 1 and c, it must contain a complete subgraph of order n_{i} whose edges are all color i. The special case above has c = 2 and n_{1} = n_{2} = 3.
Two other key theorems of Ramsey theory are:
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