Resilience of ranks of higher inclusion matrices

TL;DR

This paper proves that the rank of higher inclusion matrices remains stable under small perturbations of the family of subsets, over any field, and extends this resilience to vector space subspace incidence matrices over finite fields.

Contribution

It establishes the resilience of the rank of higher inclusion matrices over arbitrary fields and extends results to vector space subspace incidence matrices over finite fields.

Findings

Rank of higher inclusion matrices is resilient over any field.

Resilience holds when the family size is close to the full set.

Results extend to vector space subspace incidence matrices over finite fields.

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 $q$.