# On submatrices: size bound

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 $$k$$?

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 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.