Codegree conditions for (fractional) Steiner triple systems

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.

💡 Research Summary

The paper “Codegree conditions for (fractional) Steiner triple systems” by Michael Zheng advances the extremal theory of 3‑uniform hypergraphs by sharpening the minimum codegree threshold required for the existence of spanning Steiner triple systems (STS) and their fractional relaxations. A Steiner triple system on an n‑vertex set is a collection of 3‑element hyperedges such that every unordered pair of vertices belongs to exactly one hyperedge; classical necessary conditions force n ≡ 1 or 3 (mod 6). The problem studied is a hypergraph analogue of Dirac‑type results: determine how large the minimum codegree δ₂(H) (the smallest number of hyperedges containing any given pair) must be to guarantee an STS.

Lee (2023) introduced a powerful framework that reduces the existence of an STS to the existence of a perfect fractional STS, defined via a weighting ψ: E(H)→