THE EXACT TURAN NUMBER OF F(3,3) AND ALL EXTREMAL CONFIGURATIONS

If H is a 3-graph, then ex(n; H) denotes the maximum number of edges in a 3-graph on n vertices containing no sub-3-graph isomorphic to H. Let S(n) denote the 3-graph on n vertices obtained by partitioning the vertex set into parts of sizes inverted right perpendicularn(2)(n)inverted leftt perpendicular and inverted right perpendicular(2)(n)inverted left perpendicular and taking as edges all triples that intersect both parts. Let s(n) denote the number of edges in S(n). Let F(3, 3) denote the 3-graph {123, 145, 146, 156, 245, 246, 256, 345, 346, 356}. We prove that if n not equal 5, then ex(n; F(3, 3)) = s(n), and that the unique optimal 3-graph is S(n).