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Natural numbers decomposed in r disjoint subsets

Source: IMO ShortList 1990, Problem 4 (CZS 2)

January 7, 2006
number theorypartitionRamsey TheoryColoringExtremal combinatoricsIMO ShortlistHi

Problem Statement

Assume that the set of all positive integers is decomposed into r r (disjoint) subsets A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}. Prove that one of them, say Ai, A_i, has the following property: There exists a positive m m such that for any k k one can find numbers a1,a2,…,ak a_1, a_2, \ldots, a_k in Ai A_i with 0 < a_{j \plus{} 1} \minus{} a_j \leq m, (1 \leq j \leq k \minus{} 1).
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