Umeå universitets logga

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 = K − 4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that

ex2(n, K− 4 ) = n 4 + o(n).

This settles in the affirmative a conjecture of Nagle [20]. In addition, we obtain a stability result: for every near-extremal configurations 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.