Algorithmic Number On the Forehead Protocols Yielding Dense Ruzsa-Szemer\'{e}di Graphs and Hypergraphs

We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemer\'{e}di graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemer\'{e}di. The graphs induced by this protocol have $n$ vertices, $\Omega(n^2/\log n)$ edges, and are decomposable into $n^{1+O(1/\log \log n)}$ induced matchings. Another protocol is an explicit (and slightly simpler) version of the construction of Alon, Moitra and Sudakov, 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.