The codegree threshold of \(K_4^-\)

The codegree threshold \(\mathrm{ex}_2(n, F)\) 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 \(\mathrm{ex}_2(n, F)\) 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 then \(\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4\). This settles in the affirmative a conjecture of Nagle from 1999. 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 \(\mathrm{ex}_2(n, K_4^-)\) exactly and to show that tournament-based constructions \(C(T)\) are extremal for those values of \(n\).