 
Summary: Graphs and Combinatorics 4, 303306 (1988)
Graphsand
Combinatorics
© SpringerVerlag 1988
Every 8Uniform 8Regular HypergraphIs 2Colorable
N. Alon* and Z. Bregman
School of Mathematical Sciences,Sackler Faculty of Exact SciencesTel Aviv University,Ramat
Aviv,Tel Aviv,Israel
Abstract. As is wellknown, LovfiszLocal Lemma implies that every duniform dregular hyper
graph is2colorable,provided d > 9.Wepresent a differentproofofa slightlystronger result;every
duniformdregular hypergraph is 2colorable,provided d > 8.
1. Introduction
A duniform dregular hypergraph is a hypergraph in which every edge contains
precisely d vertices and every vertex is contained in precisely d edges. It is well
known (see, e.g., [4], [6]) that for d > 9 each such hypergraph is 2colorable, i.e.,
there is a 2coloring of its vertices with no monochromatic edges. To the best of
our knowledge, the only known proof of this fact is the one that applies the Lovhsz
Local Lemma to show that a random vertex coloring of the given hypergraph with
2 colors contains no monochromatic edges with positve (though very small) prob
ability. Let D be the set of all positive integers d, such that every duniform dregular
