Codegree conditions for (fractional) Steiner triple systems

TL;DR

This paper determines an upper bound on the minimum codegree needed for large 3-uniform hypergraphs to contain a Steiner triple system, improving previous bounds and linking codegree conditions to the existence of such systems.

Contribution

It establishes a tighter upper bound on codegree thresholds for fractional Steiner triple systems, advancing understanding of hypergraph conditions for Steiner systems.

Findings

Upper bound on codegree for fractional Steiner triple systems

Improved codegree threshold from previous results

Conditions ensuring Steiner triple systems in large hypergraphs

Abstract

We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in $3$-uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from Lee's paper, our results imply that if $n$ is sufficiently large and satisfies some necessary divisibility conditions, then a $3$-uniform, $n$-vertex hypergraph $H$ contains a Steiner triple system if every pair of vertices forms an edge in $H$ with at least $0.8579n$ other vertices.