Randomized trial establishes upper bound on codegree for spanning Steiner triple systems in hypergraphs, suggesting conditions for their existence.
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 [Formula: see text] is sufficiently large and satisfies some necessary divisibility conditions, then a 3-uniform, [Formula: see text]-vertex hypergraph [Formula: see text] contains a Steiner triple system if every pair of vertices forms an edge in [Formula: see text] with at least [Formula: see text] other vertices.
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Michael Zheng (2026) studied this question.
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