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

We study the Boolean rank of two families of binary matrices. The first is the binary matrix A _k,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 U _s,m of submatrices of A _k,t that is defined as U _s,m =(J _m ⊗I _s )+(I¯ _m ⊗J _s ), where I _s is the identity matrix, J _s is the all-ones matrix, s=k−2t+2 and m=(2t−2t−1). We prove that the Boolean rank of U _s,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.