Hypergraphs with the smallest Hall ratio are still hard to colour

Abstract

For $H$ a hypergraph, the Hall ratio $\rho(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{\alpha(H[X])}$ 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 $2$. Ne\v{s}et\v{r}il, R\"{o}dl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k \geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.