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In this paper, we analyze the following communication complexity problem. It is a variant of the set-disjointness problem, denoted PDISJlog N, where each of Alice and Bob gets as an input a subset of [N] of size at most log N, with the promise that the intersection of the two subsets is of size at most 1. We provide an almost tight lower bound of Â¿Â¿(log2 N) on the deterministic communication complexity of the problem. The main motivation for studying this problem comes from the so-called "clique vs. independent-set" problem, introduced by Yannakakis (1988). Proving an Â¿(log2 N) lower bound on the communication complexity of the clique vs. independent-set problem for all graphs is a long standing open problem with various implications. Proving such a lower bound for random graphs is also open. In such a graph, both the cliques and the independent sets are of size O(log N) (and obviously their intersection is of size at most 1). Hence, our Â¿Â¿(log2 N) lower bound for PDISJlog N can be viewed as a first step in this direction. Interestingly, we note that standard lower bound techniques cannot yield the desired lower bound. Hence, we develop a novel adversary argument that may find other applications.