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

Abstract

AbstractThe codegree threshold of a 3‐graph is the minimum such that every 3‐graph on vertices in which every pair of vertices is contained in at least edges contains a copy of as a subgraph. We study when , the 3‐graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that if is sufficiently large, then This settles in the affirmative a conjecture of Nagle [Congressus Numerantium, 1999, pp. 119–128]. In addition, we obtain a stability result: for every near‐extremal configuration , there is a quasirandom tournament on the same vertex set such that is ‐close in the edit distance to the 3‐graph whose edges are the cyclically oriented triangles from . For infinitely many values of , we are further able to determine exactly and to show that tournament‐based constructions are extremal for those values of .