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arXiv · 1804.07673

Tur\'an's Theorem for the Fano plane

Louis Bellmann·Christian Reiher
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Abstract

Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a complete solution to Tur\'an'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 F\"uredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.

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Louis Bellmann, Christian Reiher. 2018-04-20. Tur\'an's Theorem for the Fano plane. https://doi.org/10.1007/s00493-019-3981-8

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