The Turán Number Of The Fano Plane

Let PG ₂(2) be the Fano plane, i. e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3-uniform hypergraph on n vertices not containing a Fano plane is

$$ ex{\left( {n,PG_{2} {\left( 2 \right)}} \right)} = {\left( {\begin{array}{*{20}c} {n} \\ {3} \\ \end{array} } \right)} - {\left( {\begin{array}{*{20}c} {{{\left\lfloor {n/2} \right\rfloor }}} \\ {3} \\ \end{array} } \right)} - {\left( {\begin{array}{*{20}c} {{{\left\lceil {n/2} \right\rceil }}} \\ {3} \\ \end{array} } \right)}. $$

Moreover, the only extremal configuration can be obtained by partitioning an n-element set into two almost equal parts, and taking all the triples that intersect both of them. This extends an earlier result of de Caen and Füredi, and proves an old conjecture of V. Sós. In addition, we also prove a stability result for the Fano plane, which says that a 3-uniform hypergraph with density close to 3/4 and no Fano plane is approximately 2-colorable.