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

Alon, Frankl, and Lov\'asz proved a conjecture of Erd\H{o}s that one needs at least $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of $\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same color are not pairwise disjoint. A generalization of this problem where one requires $s$-wise instead of pairwise intersections was considered by Sarkaria. He claimed a proof of a generalized Erd\H{o}s--Kneser conjecture establishing a lower bound for the number of colors that reduces to Erd\H{o}s' original conjecture for ${s = 2}$. Lange and Ziegler pointed out that his proof fails whenever $r$ is not a prime. Here we establish this generalized Erd\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 Abyazi Sani and Alishahi, and prove the latter in several cases.