# On the Codegree Density of Complete 3-Graphs and Related Problems

### Abstract

Given a 3-graph $F$, its codegree threshold $\textrm{co-ex}(n, F)$ is 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 $d$ triples but which contains no member of $F$ as a subgraph. The limit

\[\gamma(F)=\lim_{n\rightarrow \infty} \frac{\textrm{co-ex}(n,F)}{n-2}\]

is known to exist and is called the codegree density of $F$.

In this paper we generalise a construction of Czygrinow and Nagle to bound below the codegree density of complete 3-graphs: for all integers $s\geq 4$, the codegree density of the complete 3-graph on $s$ vertices $K_s$ satisfies

\[\gamma(K_s)\geq 1-\frac{1}{s-2}.\]

We then provide constructions based on Steiner triple systems which show that if this lower bound is sharp, then we do not have stability in general.

In addition we prove bounds on the codegree density for two other infinite families of 3-graphs.