The codegree threshold of K^- ₄

The codegree threshold ex(2)(n, F) of a 3-graph F is the minimum d = d(n) such that every 3-graph on n vertices in which every pair of vertices is contained in at least d + 1 edges contains a copy of F as a subgraph. We study ex(2)(n, F) when F = K-4(-) , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that if n is sufficiently large, thenex(2)(n, K-4(-))<= n + 1/ 4 .This settles in the affirmative a conjecture of Nagle [Congressus Numerantium, 1999, pp. 119-128]. In addition, we obtain a stability result: for every near-extremal configuration G, there is a quasirandom tournament T on the same vertex set such that G is o(n3)-close in the edit distance to the 3-graph C(T) whose edges are the cyclically oriented triangles from T. For infinitely many values of n, we are further able to determine ex(2)(n, K-4(-)) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.