The codegree threshold of $K_4^-$

The codegree threshold ex2(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 ex2(n, F ) when F = K− 4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that ex2(n,K − 4 ) ≤ n+ 1 4 . This settles in the affirmative a conjecture of Nagle [25]. 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,K 4 ) exactly and to show that tournament-based constructions C(T ) are extremal for those values of n.

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