We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa–Szemeredi graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemeredi. 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 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.

中文翻译：

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额头协议上的数字产生密集的 Ruzsa-Szemerédi 图和超图

我们描述了提供密集 Ruzsa-Szemeredi 图的额头协议的算法数字。一种协议导致对 Ruzsa 和 Szemeredi 的原始构造进行简单而自然的扩展。由该协议导出的图有 n 个顶点，$$\Omega(n^2/\log n)$$ 条边，可分解为 $$n^{1+O(1/\log \log n)}$ $诱导匹配。另一种协议是 [1] 结构的一个稍微简单的版本，生成具有相似属性的图。我们还将上述协议推广到三个以上的玩家，以构建密集的均匀超图，其中每条边都位于少数正单形中，扩展了 Fox 和 Loh 的结果。