On the Codegree Density of Complete 3-Graphs and Related Problems (original) (raw)
Skip to main content Skip to main navigation menu Skip to site footer
Keywords: Extremal hypergraph theory, codegree density
Abstract
Given a 3-graph FFF, its codegree threshold textrmco−ex(n,F)\textrm{co-ex}(n, F)textrmco−ex(n,F) is the largest number d=d(n)d=d(n)d=d(n) such that there exists an nnn-vertex 3-graph in which every pair of vertices is contained in at least ddd triples but which contains no member of FFF 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 FFF.
In this paper we generalise a construction of Czygrinow and Nagle to bound below the codegree density of complete 3-graphs: for all integers sgeq4s\geq 4sgeq4, the codegree density of the complete 3-graph on sss vertices KsK_sKs 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.
How to Cite
Falgas-Ravry, V. (2013). On the Codegree Density of Complete 3-Graphs and Related Problems. The Electronic Journal of Combinatorics, 20(4), #P28. https://doi.org/10.37236/3578