Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

Abstract

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ 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 Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž 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.