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Concept
Ramsey's theorem
Summary
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 r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)
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, c, and any given integers n_1, ..., n_c, there is a number, R(n_1, ..., n_c), such that if the edges of a complete graph of order R(n_1, ..., n_c) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order n_i whose edges are all colour i. The special case above has c = 2 (and n_1 = r and n_2 = s).
Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex, v. There are 5 edges incident to v 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, v, to vertices, r, s and t, are blue. (If not, exchange red and blue in what follows.) If any of the edges, (rs), (rt), (st), are also blue then we have an entirely blue triangle.
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