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Alon, Frankl, and Lov\'asz proved a conjecture of Erd\H{o}s that one needs atleast $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of$\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same colorare not pairwise disjoint. A generalization of this problem where one requires$s$-wise instead of pairwise intersections was considered by Sarkaria. Heclaimed a proof of a generalized Erd\H{o}s--Kneser conjecture establishing alower bound for the number of colors that reduces to Erd\H{o}s' originalconjecture for ${s = 2}$. Lange and Ziegler pointed out that his proof failswhenever $r$ is not a prime. Here we establish this generalizedErd\H{o}s--Kneser conjecture for every $r$, as long as $s$ is not too close to$r$. Our result encompasses earlier results but is significantly more general.We discuss relations of our results to conjectures of Ziegler and of AbyaziSani and Alishahi, and prove the latter in several cases.

On the generalized Erd\H{o}s--Kneser conjecture: proofs and reductions