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We consider combinatorial properties of Boolean matrices and their application to two-party communication complexity. Let A be a binary n×n matrix and let K be a field. Rectangles are sets of entries defined by collections of rows and columns. We denote by rankB(A) (rankK(A), resp.) the least size of a family of rectangles whose union (sum, resp.) equals A. We prove the following: - With probability approaching 1, for a random Boolean matrix A the following holds: rankB(A) ≥ n(1-o(1)). - For finite K and fixed Ɛ 0 the following holds: If A is a Boolean matrix with rankB(A) ≥ t then there is some matrix A′ ≥ A such that A-A′ has at most Ɛċn2 non-zero entries and rankK(A′) ≥ tO(1)(1). As applications we mention some improvements of earlier results: (1) With probability approaching 1 a random n-variable Boolean function has nondeterministic communication complexity n, (2) functions with nondeterministic communication complexity l can be approximated by functions with parity communication complexity O(l). The latter complements a result saying that nondeterministic and parity communication protocols cannot efficiently simulate each other. Another consequence is: (3) matrices with small Boolean rank have small matrix rigidity over any field.