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arXiv · 2503.23252

Steiner triple systems with high discrepancy

Lior Gishboliner·Stefan Glock·Amedeo Sgueglia
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Abstract

In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple system of order $n$ with discrepancy $\Omega(n^2)$. This is not true for $r=2$, but we are able to asymptotically characterise all $2$-colourings which do not contain a Steiner triple system with high discrepancy. The key step in our proofs is a characterization of 3-uniform hypergraphs avoiding a certain natural type of induced subgraphs, contributing to the structural theory of hypergraphs.

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BibTeXRIS

Lior Gishboliner, Stefan Glock, Amedeo Sgueglia. 2025-03-29. Steiner triple systems with high discrepancy. https://arxiv.org/abs/2503.23252

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