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

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.