The codegree threshold of K4−

The codegree threshold ex2(n, F) of a non-empty 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 ex2(n, F) when F=K4−, the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove thatex2(n,K4−)=n4+O(1). This settles in the affirmative a conjecture of Nagle [Nagle, B., Turán-Related Problems for Hypergraphs, Congr. Numer. (1999), 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 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 ex2(n,K4−) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.