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Turán's Theorem for the Fano plane

Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ 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 Füredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.