Ramsey's theorem

High Quality Content by WIKIPEDIA articles! In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph (that is, a simple graph in which an edge connects every pair of vertices), one will find monochromatic complete subgraphs. For 2 colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.

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High Quality Content by WIKIPEDIA articles! In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph (that is, a simple graph in which an edge connects every pair of vertices), one will find monochromatic complete subgraphs. For 2 colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.