When $t$-intersecting hypergraphs admit bounded $c$-strong colourings

TL;DR

This paper proves that every t-intersecting hypergraph has a bounded (t+1)-strong chromatic number, resolving an open problem, and characterizes when larger c-strong chromatic numbers occur, also relating to sunflower-free hypergraphs.

Contribution

It establishes that t-intersecting hypergraphs have bounded (t+1)-strong chromatic number and characterizes conditions for larger c-strong chromatic numbers, extending to sunflower-free hypergraphs.

Findings

Every t-intersecting hypergraph has bounded (t+1)-strong chromatic number.

Characterization of when t-intersecting hypergraphs have large c-strong chromatic number for c ≥ t+2.

Application of results to hypergraphs excluding sunflowers with specific parameters.

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.