Question 1
(BUL 1) The set $M=\{1,2,\ldots,2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $n\ge k^3+k$. Prove that there exist even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ ($1\le i\le k$) such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_j$ ($1\le j\le k$).