The Codegree Threshold for 3-Graphs with Independent Neighborhoods

Given a family of 3-graphs F , we define its codegree threshold coex(n,F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe and cde. In this paper, we give two proofs that coex(n, {F3,2}) = ( 1 3 + o(1) ) n, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large n: coex(n, {F3,2}) = { bn/3c − 1 if n is congruent to 1 modulo 3 bn/3c otherwise. In addition we determine the set of codegree-extremal configurations.