 
Summary: Probabilistic Methods in Coloring and Decomposition Problems
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
Numerous problems in Graph Theory and Combinatorics can be formulated in terms of the
existence of certain colorings of graphs or hypergraphs. Many of these problems can be solved or
partially solved by applying probabilistic arguments. In this paper we discuss several examples
that illustrate the methods used. This is mainly a survey paper, but it contains some new results
as well.
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1 Probability and Coloring; Older Examples
Probabilistic methods have been useful in combinatorics for almost fifty years. Many examples
dealing with various branches of Combinatorics can be found in [20] and in [31]. Coloring is one of
the most popular areas in Combinatorics and in Graph Theory and has been the source of many
intriguing problems for decades. It is therfore not surprising that there are various rather old known
applications of probabilistic techniques in different coloring problems. In this section we discuss
briefly some representing examples.
Recall that the Ramsey number r(k, l) is the smallest integer r such that in any 2coloring of
