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 $k$?
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On submatrices: size bound
To each such matrix $A$ we can associate $A\_1,A\_2$, the sets of column indices and row indices respectively. The families $\mathcal F\_i=\{A\_i| A\in \mathcal F\}$ are intersecting families of subsets, therefore by [Erdos-Ko-Rado](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem) we have
$$|\mathcal F\_i|\le \binom{2n-1}{k-1}$$
Since a matrix $A$ is uniquely determined by its set of rows and columns we have
$$|\mathcal F|\le |\mathcal F\_1||\mathcal F\_2|\le \binom{2n-1}{k-1}^2$$
and this maximum can be achieved by taking all the $k\times k$ submatrices that contain the $(1,1)$ entry.


