On the codegree density of complete 3-graphs and related problems

TL;DR

This paper investigates the codegree density of complete 3-graphs and related structures, providing new lower bounds, constructions, and bounds for various families of 3-graphs, advancing understanding in extremal hypergraph theory.

Contribution

It generalizes previous constructions to establish lower bounds on codegree density for complete 3-graphs and introduces Steiner triple system-based constructions to explore stability issues.

Findings

Lower bound gamma(K_s) ≥ 1 - 1/(s-2) for complete 3-graphs

Construction methods using Steiner triple systems

Bounds on codegree density for other 3-graph families

Abstract

Given a family of 3-graphs F its codegree threshold coex(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 3-edges but which contains no member of F as a subgraph. The codegree density gamma(F) is the limit of coex(n,F)/(n-2) as n tends to infinity. 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>3, the codegree density of the complete 3-graph on s vertices K_s satisfies gamma(K_s)\geq 1-1/(s-2). We also 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.