On the generalized Erdős--Kneser conjecture: proofs and reductions

Abstract

Alon, Frankl, and Lovász proved a conjecture of Erdős 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 Erdős--Kneser conjecture establishing a lower bound for the number of colors that reduces to Erdős' 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 Erdős--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.

arXiv comment8 pages