The Exact Turán 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 ⌈ n 2 ⌉ and ⌊ n 2 ⌋ 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 6= 5 then ex(n;F (3, 3)) = s(n) and that the unique optimal 3-graph is S(n).