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

TL;DR

This paper proves a generalized Erdős–Kneser conjecture for all values of r when s is not too close to r, extending previous results and addressing gaps in earlier proofs, with implications for related conjectures.

Contribution

The paper establishes the generalized Erdős–Kneser conjecture for all r when s is not too close to r, overcoming limitations of previous proofs.

Findings

Proved the generalized Erdős–Kneser conjecture for all r with s not close to r.

Extended earlier results to a more general setting.

Connected results to conjectures by Ziegler and Abyazi Sani and Alishahi.

Abstract

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.