On the rank of higher inclusion matrices

TL;DR

This paper investigates the maximum size of r-graphs with constrained rank of their higher inclusion matrices, providing exact results for certain parameters and answering a question posed by Keevash.

Contribution

It defines and determines the rank-extremal function for higher inclusion matrices in r-graphs, extending previous work and solving a specific open problem.

Findings

Exact determination of the rank-extremal function for t linear in n

Identification of extremal r-graphs with constrained matrix rank

Resolution of a question posed by Keevash regarding the case t=1

Abstract

Let r >= s >= 0 be integers and G be an r-graph. The higher inclusion matrix M_s^r(G) is a {0,1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V(G) of size s: the entry corresponding to an edge e and a subset S is 1 if S is contained in e and 0 otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function rex(n,t,r,s) as the maximum number of edges of an r-graph G having rank M_s^r(G) <=\binom{n}{s} - t. For t at most linear in n we determine this function as well as the extremal r-graphs. The special case t=1 answers a question of Keevash.