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 $q$.
Download