An Lp version of the beck-fiala conjecture

Abstract

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 : X → {-1, +1} exists such that |χ(S)| = O(√t) holds for all S ∈ S, where χ (S) = Σχ∈S χ (χ). We prove a weaker statement, namely that for any fixed p ≥ 1, a coloring X 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). © 1998 Academic Press Limited.