Number on the Forehead Protocols yielding dense Ruzsa–Szemerédi graphs and hypergraphs

We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa–Szemerédi graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemerédi. The graphs induced by this protocol have n vertices, Ω(n^2/log n) edges, and are decomposable into n^1+O(1/loglog n) induced matchings. Another protocol is a somewhat simpler version of the construction of [1], producing graphs with similar properties. We also generalize the above protocols to more than three players, in order to construct dense uniform hypergraphs in which every edge lies in a positive small number of simplices, extending a result of Fox and Loh.