On the chromatic number of generalized Kneser hypergraphs

TL;DR

This paper establishes a new lower bound on the chromatic number of generalized Kneser hypergraphs, improving previous results and advancing understanding of their coloring properties.

Contribution

It provides an improved lower bound for the chromatic number of generalized Kneser hypergraphs under certain conditions, extending prior work by Alon, Frankl, and Lovász.

Findings

New lower bound for chromatic number derived

Bound improves upon previous results by Alon--Frankl--Lovász

Applicable for specified parameter ranges in hypergraph coloring

Abstract

The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their intersection. In this note, we show that for each non-negative integers $k, n, r, s$ satisfying $n \geq r(k-1)+1$, $k > s\geq 0$, and $r\geq 2$, we have $$\chi ({KG}^{r}(n,k,s))\geq\left\lceil\frac{n-r(k-s-1)}{r-1}\right\rceil,$$ which improves the previously known result by Alon--Frankl--Lov\'{a}sz.