On the properties of a family of submatrices Let $n$ and $k \le n/2$ be positive integers, and consider a family $\mathcal{F}$ of $k\times k$ sub-matrices of an $n\times n$ matrix such that every two elements of $\mathcal{F}$ intersect. What non-trivial (I know I'm already annoying people by this term) properties of $\mathcal{F}$ one could claim? (in particular related to its cardinality).
As @Steven Stadnicki points out in his comment, acording to your question, the contents of the matrices are not important here, but the number $k$ and $n$. Therefore, you can look at $\mathcal{F}$ as a family of intersecting $k$-subsets of $\{1,\dots,n\}$.

You can derive properties from something called a Sunflower.

Definition: A sunflower with $m$ petals and a core $Y$ is a collection of sets $S_1,\dots,S_m$ such that $S_i\cap S_j=Y$ for all $i\neq j$. Moreover, the sets $S_i\setminus Y\neq \emptyset$.

Sunflower Lemma: Let $\mathcal{F}$ be a family of sets each of cardinality $k$. If $|\mathcal{F}|&gt;k!(k-1)^k$ then $\mathcal{F}$ contains a sunflower with $k$ petals.

You can make relaxations to the sunflower, like the petals not being intersected by a set of size smaller than $c$. In this case we have the following lemma.

Lemma: Let $\mathcal{F}$ be a family of sets each of cardinality $k$. If $|\mathcal{F}| &gt; (c-1)^k$ then $\mathcal{F}$ contains a flower with $k$ petals.

There are more properties that you can look for in the book:

Extremal Combinatorics with Applications to Computer Science by Stasys Jukna.

The definitions and lemmas above are from chapter 7. Take a look at the excercises, and maybe also chapter 8 on Intersecting Families.