  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.
