✦ LIBER ✦

AnLpVersion of the Beck–Fiala Conjecture

✍ Scribed by J Matoušek

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 139 KB
Volume: 19
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.1997.0162

No coin nor oath required. For personal study only.

✦ Synopsis

Beck and Fiala conjectured in 1981 that for any set system S of maximum degree t on a finite ground set X , a coloring χ :

holds for all S ∈ S, where χ(S) = x∈S χ(x). We prove a weaker statement, namely that for any fixed p ≥ 1, a coloring χ exists such that the pth degree average of |χ(S)| over S ∈ S is O( √ t). The result also holds if each set is assigned a nonnegative real weight and the pth degree average is taken with these weights (with χ depending on the weights).

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