Chromatic numbers of stable Kneser hypergraphs via topological Tverberg-type theorems

TL;DR

This paper generalizes classical Kneser conjecture results to stable Kneser hypergraphs, using topological Tverberg theorems to establish new lower bounds on their chromatic numbers.

Contribution

It introduces a unified framework that extends multiple prior results and develops a topological approach to bound chromatic numbers of stable Kneser hypergraphs.

Findings

Determined chromatic numbers of certain sparse stable Kneser hypergraphs.

Developed a method combining equivariant topology and convex hull intersection results.

Provided new lower bounds for hypergraph chromatic numbers.

Abstract

Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lov\'asz proved a different generalization of Kneser's conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lov\'asz' result to arbitrary set systems, while K\v{r}\'{i}\v{z} did the same for the $r$-fold extension of Kneser's conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.