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- W4295005029 abstract "Alon, Frankl, and Lov'asz proved a conjecture of ErdH{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 ErdH{o}s--Kneser conjecture establishing a lower bound for the number of colors that reduces to ErdH{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 ErdH{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." @default.
- W4295005029 created "2022-09-09" @default.
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- W4295005029 date "2017-12-09" @default.
- W4295005029 modified "2023-10-18" @default.
- W4295005029 title "On the generalized ErdH{o}s--Kneser conjecture: proofs and reductions" @default.
- W4295005029 doi "https://doi.org/10.48550/arxiv.1712.03456" @default.
- W4295005029 hasPublicationYear "2017" @default.
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