  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.
