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partition of N in 3 subsets, a_1 < a_2 < ... < a_k and a_{j+1} -a_j \le m

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1991 p6

February 11, 2020
partitionSubsetscombinatorics

Problem Statement

The set NN is partitioned into three subsets A1,A2,A3A_1,A_2,A_3. Prove that at least one of them has the following property: There exists a positive number mm such that for any kk one can find numbers a1<a2<...<aka_1 < a_2 < ... < a_k in that subset satisfying aj+1−aj≤ma_{j+1} -a_j \le m for j=1,...,k−1j = 1,...,k -1.
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