Probabilistic Methods in Coloring and Decomposition Problems Department of Mathematics 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. 0 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 2-coloring of Collections: Mathematics