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 / log log 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.