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partition [2n] into k disjoint subsets, existence of even numbers in subset

Source: Bulgaria 1979 P6

June 17, 2021
number theory

Problem Statement

The set M={1,2,…,2n} (n≥2)M=\{1,2,\ldots,2n\}~(n\ge2) is partitioned into kk nonintersecting subsets M1,M2,…,MkM_1,M_2,\ldots,M_k, where k3+1≤nk^3+1\le n. Prove that there exist k+1k+1 even numbers 2j1,2j2,…,2jk+12j_1,2j_2,\ldots,2j_{k+1} in MM that are in one and the same subset MjM_j (1≤j≤k)(1\le j\le k) such that the numbers 2j1−1,2j2−1,…,2jk+1−12j_1-1,2j_2-1,\ldots,2j_{k+1}-1 are also in one and the same subset MrM_r (1≤r≤k)(1\le r\le k).
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