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arXiv · 2610.20562

Hypergraphs with the smallest Hall ratio are still hard to colour

Veronica Bitonti·Freddie Illingworth·Pedram Noohi·Jane Tan
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Abstract

For $H$ a hypergraph, the Hall ratio $ρ(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{α(H[X])}$ 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 $k \geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.

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Veronica Bitonti, Freddie Illingworth, Pedram Noohi, Jane Tan. 2026-10-14. Hypergraphs with the smallest Hall ratio are still hard to colour. https://arxiv.org/abs/2610.20562

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