Dependent Random Graphs and Multi-Party Pointer Jumping ˚

We initiate a study of a relaxed version of the standard Erdős-Rényi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs. Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size p1´op1qq logpnqlogp1{pq, and the chromatic number will be at most n logp1{p1´pqqlogn. We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping (mpjk) problem in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for mpj3 costs Opnplog lognq { lognq communication, improving on a bound from [9]. We extend our protocol to the non-Boolean pointer jumping problem ympjk, achieving an upper bound which is opnq for any k ě 4 players. This is the first opnq protocol for ympjk and improves on a bound of Damm, Jukna, and Sgall [12], which has stood for almost twenty years