The codegree threshold of $K_4^-$

TL;DR

This paper determines the minimum codegree threshold for containing a specific 3-graph $K_4^-$, confirming a longstanding conjecture, and characterizes near-extremal configurations using flag algebra methods and stability analysis.

Contribution

It proves the conjectured upper bound for large $n$, establishes stability results, and exactly determines the threshold for infinitely many $n$, advancing understanding of 3-graph extremal problems.

Findings

Proved $ ext{ex}_2(n, K_4^-) extleq (n+1)/4$ for large $n$.

Established stability: near-extremal graphs are close to cyclically oriented triangle constructions.

Determined exact thresholds for infinitely many $n$, showing extremality of tournament-based constructions.

Abstract

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 $n$.