The codegree threshold of K4−$K_4^{-}$

The codegree threshold ex 2 ( n , F ) $\operatorname{ex}_2(n, F)$ of a 3-graph $F$ is the minimum d = $d=d(n)$ such that every on $n$ vertices in which pair contained at least + 1 $d+1$ edges contains copy as subgraph. We study when K 4 − $F=K_4^-$ with 3 edges. Using flag algebra techniques, we prove if sufficiently large, then ⩽ . $$\begin{equation*} \hspace*{76pt}\operatorname{ex}_2(n, K_4^-)\leqslant \frac{n+1}{4}.\hspace*{-76pt} \end{equation*}$$ This settles affirmative conjecture Nagle [Congressus Numerantium, 1999, pp. 119–128]. In addition, obtain stability result: for near-extremal configuration G $G$ there quasirandom tournament T $T$ same vertex set o $o(n^3)$ -close edit distance to C $C(T)$ whose are cyclically oriented triangles from For infinitely many values further able determine K_4^-)$ exactly and show tournament-based constructions extremal those