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A four-coloring of the set M (IMO SL 1987-P17)

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August 19, 2010
arithmetic sequencecombinatoricsProbabilistic MethodColoringExtremal combinatoricsIMO Shortlist

Problem Statement

Prove that there exists a four-coloring of the set M={1,2,⋯ ,1987}M = \{1, 2, \cdots, 1987\} such that any arithmetic progression with 1010 terms in the set MM is not monochromatic.
Alternative formulation
Let M={1,2,⋯ ,1987}M = \{1, 2, \cdots, 1987\}. Prove that there is a function f:M→{1,2,3,4}f : M \to \{1, 2, 3, 4\} that is not constant on every set of 1010 terms from MM that form an arithmetic progression.
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