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Find the least h(r)

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September 14, 2010
inequalitiesnumber theory unsolvednumber theory

Problem Statement

For every integer r>1r > 1 find the smallest integer h(r)>1h(r) > 1 having the following property: For any partition of the set {1,2,...,h(r)}\{1, 2, . . ., h(r)\} into rr classes, there exist integers a≥0,1≤x≤ya \geq 0, 1 \leq x \leq y such that the numbers a+x,a+y,a+x+ya + x, a + y, a + x + y are contained in the same class of the partition.
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