The codegree threshold for 3-graphs with independent neighbourhoods

Abstract

Given a family of 3-graphs $F$, we define its codegree threshold $\mathrm{coex}(n,F)$ to be the largest number $d=d(n)$ such that there exists an $n$-vertex 3-graph in which every pair of vertices is contained in at least $d$ 3-edges but which contains no member of $F$ as a subgraph. Let $F_{3,2}$ be the 3-graph on $\{a,b,c,d,e\}$ with 3-edges $\{abc,abd,abe,cde\}$. In this paper, we give two proofs that $\mathrm{coex}(n,F_{3,2})=n/3+o(n)$, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large $n$: $\mathrm{coex}(n,F_{3,2})=\lfloor n/3\rfloor -1$ if $n$ is congruent to $1$ modulo $3$, and $\lfloor n/3\rfloor$ otherwise. In addition we determine the set of codegree-extremal configurations.

40 pages, 2 figures, 3 ancillary files

Cited by in corpus (1)

• Minimum Number of k-Cliques in Graphs with Bounded Independence Number