Question 46
Let $s \ge 2$ and $n \ge k \ge 2$ be integers, and let $\mathcal{A}$ be a subset of $\{1, 2, \dots, n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $\mathcal{A}$ share some entry. Prove that there are an integer $p \le k$ and $s+2$ members $A_1, A_2, \dots, A_{s+2}$ of $\mathcal{A}$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i \ne j$.
Miroslav Mironov, Bulgaria
Miroslav Mironov, Bulgaria