# A new lower bound for the chromatic number of general Kneser hypergraphs

## TL;DR

This paper introduces a new combinatorial parameter called equitable colorability defect that improves lower bounds for the chromatic number of general Kneser hypergraphs, surpassing previous bounds and ensuring the existence of colorful subhypergraphs.

## Contribution

It defines the equitable colorability defect and establishes a new, stronger lower bound for the chromatic number of general Kneser hypergraphs, improving upon prior bounds by Ziegler and Dol'nikov-Kríz.

## Key findings

- The new lower bound always matches or exceeds Ziegler's bound.
- Several hypergraph families show the new bound can be arbitrarily larger than previous bounds.
- The existence of colorful subhypergraphs in any proper coloring is guaranteed.

## Abstract

A general Kneser hypergraph ${\rm KG}^r(\mathcal{H})$ is an $r$-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph $\mathcal{H}$. The colorability defect of $\mathcal{H}$ is a combinatorial parameter providing a lower bound for the chromatic number of ${\rm KG}^r(\mathcal{H})$ which is addressed in a series of works by Dol'nikov [Sibirskii Matematicheskii Zhurnal, 1988}], K\v{r}\'{\i}\v{z} [Transaction of the American Mathematical Society, 1992], and Ziegler~[Inventiones Mathematicae, 2002]. In this paper, we define a new combinatorial parameter, the equitable colorability defect of hypergraphs, which provides some common improvements of these works. Roughly speaking, we propose a new lower bound for the chromatic number of general Kneser hypergraphs which substantially improves Ziegler's lower bound. It is always as good as Ziegler's lower bound and we provide several families of hypergraphs for which the difference between these two lower bounds is arbitrary large. This specializes to a substantial improvement of the Dol'nikov-K\v{r}\'{\i}\v{z} lower bound for the chromatic number of general Kneser hypergraphs as well. Furthermore, we prove a result ensuring the existence of a colorful subhypergraph in any proper coloring of general Kneser hypergraphs which strengthens Meunier's result [The Electronic Journal of Combinatorics, 2014].