The Boolean rank of the uniform intersection matrix and a family of its submatrices

Abstract

We study the Boolean rank of two families of binary matrices. The first is the binary matrix Ak,t that represents the adjacency matrix of the intersection bipartite graph of all subsets of size t of {1,2,...,k}. We prove that its Boolean rank is k for every k≥2t.The second family is the family Us,m of submatrices of Ak,t that is defined as Us,m=(Jm⊗Is)+(I¯m⊗Js), where Is is the identity matrix, Js is the all-ones matrix, s=k−2t+2 and m=(2t−2t−1). We prove that the Boolean rank of Us,m is also k for the following values of t and s: for s=2 and any t≥2, that is k=2t; for t=3 and any s≥2; and for any t≥2 and s>2t−2, that is k>4t−4.