Steiner triple systems with high discrepancy

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 $n\equiv 1,3(\mathrm{mod}6)$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple system of order $n$ with discrepancy $\hat{I}\text{c◯}(n^2)$. 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.