Public vs. private coin flips in one round communication games

We study 1-round two parties communication complexity games, where private random bits are used. We observe that the existence of good protocols for such games is related to a notion of approximating the matrix that represents the function by a certain low rank matrix. This gives rise to a new notion of rank, analogous of positive rank for 1-round communication complexity. We prove that the identity matrix is non approximable by low rank matrices in this sense. As a corollary we prove that any randomized protocol for the equality function requires Ω(√n) complexity in the 1-round, private bits model, answering an open question raised by Yao [8]. A corollary is an answer to the following graph theoretic question: Assume a graph on n vertices has a set of N = N(n) cliques and so that for each two different cliques of this set, the number of edges between them is at most 0.1 of the product of their sizes. How large can N be ? We show that N = n^{θ(log n)} is the right of magnitude.