Hypergraphs with the smallest Hall ratio are still hard to colour

Abstract

For $H$ a hypergraph, the Hall ratio $\rho (H)={\text{max}}_{X\subseteq V(H)}\frac{|X|}{\alpha (H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k=2$), having minimum Hall ratio $2$ forces chromatic number $2$. Nešetřil, Rödl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k⩾3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.