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Determine the smallest h(r) (IMO SL 1987-P18)

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August 19, 2010
combinatoricspartitionColoringRamsey TheorySet systemsIMO Shortlist

Problem Statement

For any integer r≥1r \geq 1, determine the smallest integer h(r)≥1h(r) \geq 1 such that for any partition of the set {1,2,⋯ ,h(r)}\{1, 2, \cdots, h(r)\} into rr classes, there are integers a≥0 ;1≤x≤ya \geq 0 \ ; 1 \leq x \leq y, such that a+x,a+y,a+x+ya + x, a + y, a + x + y belong to the same class.
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