When \(t\)-intersecting hypergraphs admit bounded \(c\)-strong colourings

Abstract

The \(c\)-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least \(c\) colours or is rainbow. We show that every \(t\)-intersecting hypergraph has bounded \((t + 1)\)-strong chromatic number, resolving a problem of Blais, Weinstein and Yoshida. In fact, we characterise when a \(t\)-intersecting hypergraph has large \(c\)-strong chromatic number for \(c\geq t+2\). Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters.