Let $M$ be a generic $2n\times 2n$ matrix and fix $k\leq n$.

Suppose $\mathcal{F}$ is a family of submatrices under the conditions that $A\in\mathcal{F}$ provided

(a) $A$ is a $k\times k$ submatrix of $M$, and

(b) $A$ overlaps with each and every $B\in\mathcal{F}$.

QUESTION.What is the best upper bound for the cardinality of $\mathcal{F}$, in terms of $n$ and $k$?