 
Summary: How to color shift hypergraphs
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Igor Kriz
Department of Mathematics
University of Chicago, Chicago, IL 60637, USA
and Jaroslav Nesetril
Department of Applied Mathematics
Charles University, 11800 Praha 1, Czechoslovakia
Abstract
Let g(k) denote the minimum integer m so that for every set S of m integers there is a
kcoloring of the set of all integers so that every translate of S meets every color class. It is a
well known consequence of the Local Lemma that g(k) is finite for all k. Here we present a new
proof for this fact, that yields a very efficient parallel algorithm for finding, for a given set S, a
coloring as above. We also discuss the problem of finding colorings so that every translate of S
has about the same number of points in each color. In addition, we prove that for large k
(1 + o(1))k log k g(k) (3 + o(1))k log k.
