When t-intersecting hypergraphs admit bounded c-strong colourings

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≥ t+2. Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters.