Confirming a conjecture of Vera T. Sos in a very strong sense, we give a complete solution to Turan's hypergraph problem for the Fano plane. That is we prove for n≥8 that among all 3-uniform hypergraphs on n vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for n = 7 there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order 7 by removing all five edges containing a fixed pair of vertices. For sufficiently large values n this was proved earlier by Furedi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.

中文翻译：

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法诺平面的图兰定理 Louis Bellmann, Christian Reiher

在很强的意义上证实了 Vera T. Sos 的一个猜想，我们给出了 Fano 平面图兰超图问题的完整解。也就是说，我们证明对于 n≥8，在不包含 Fano 平面的 n 个顶点上的所有 3-均匀超图中确实有一个边数最大的超图，即平衡的、完全的、二分超图。此外，对于 n = 7，正好有另一个具有相同边数的极值配置：通过删除包含一对固定顶点的所有五个边，从 7 阶团产生的超图。对于足够大的 n 值，Furedi 和 Simonovits 以及使用稳定性方法的 Keevash 和 Sudakov 较早地证明了这一点。