Resilience of ranks of higher inclusion matrices

Abstract

Abstract. Let n ≥ r ≥ s ≥ 0 be integers and F a family of r-subsets of [n]. Let W F r,s be the higher inclusion matrix of the subsets in F vs. the s-subsets of [n]. When F consists of all r-subsets of [n], we shall simply write W r,s in place of W F r,s . 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 F is "close" to n r then rank K (W F r,s ) = 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 F q is also resilient if char(K) is coprime to q.

A representation-theoretic computation of the rank of $1$-intersection incidence matrices: $2$-subsets vs. $n$-subsets

“…The study of inclusion matrices has applications to the theory of designs. For much information and more history of these matrices see [18,Section 10] and [15].…”