Resilience of ranks of higher inclusion matrices

Abstract

Let $n \geq r \geq s \geq 0$ be integers and $\mathcal{F}$ a family of $r$-subsets of $[n]$. Let $W_{r,s}^{\mathcal{F}}$ be the higher inclusion matrix of the subsets in ${\mathcal F}$ vs. the $s$-subsets of $[n]$. When $\mathcal{F}$ consists of all $r$-subsets of $[n]$, we shall simply write $W_{r,s}$ in place of $W_{r,s}^{\mathcal{F}}$. In this paper we prove that the rank of the higher inclusion matrix $W_{r,s}$ over an arbitrary field $K$ is resilient. That is, if the size of $\mathcal{F}$ is "close" to ${n \choose r}$ then $\mbox{rank}_{K}(W_{r,s}^{\mathcal{F}}) = \mbox{rank}_{K}(W_{r,s})$, where $K$ is an arbitrary field. Furthermore, we prove that the rank (over a field $K$) of the higher inclusion matrix of $r$-subspaces vs. $s$-subspaces of an $n$-dimensional vector space over $\mathbb{F}_q$ is also resilient if ${\rm char}(K)$ is coprime to $q$.