The Maximum Size of 3-Uniform Hypergraphs Not Containing a Fano Plane

A conjecture of V. Sós [3] is proved that any set of 34 (n 3 ) + cn2 triples from an n-set, where c is a suitable absolute constant, must contain a copy of the Fano configuration (the projective plane of order two). This is an asymptotically sharp estimate. Given a 3-uniform hypergraph F , let ex3(n,F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain any subhypergraph isomorphic to F . Our terminology follows that of [1], which is a comprehensive survey of Turántype extremal problems. An elementary and well known averaging argument shows that the ratio ex3(n,F)/ ( n 3 ) is a non-increasing sequence, so that π(F) := lim n→∞ ex3(n,F)/ ( n 3 )