Random Graphs

Abstract

“Give a ‘good’ lower bound on the Ramsey number R(s, s), that is show that there exists a graph of large order such that neither the graph nor its complement contains a K ^s”. “Show that for every natural number k there is a k-chromatic graph which does not contain a triangle”. It does not take long to realize that the constructions that seem to be demanded by questions like these are not easily come by. Later we show that for every k there is a graph with the latter property, but even for k = 4 our graph has at least 2³² vertices. This book does not contain a picture of such a graph. Indeed, the aim of this chapter is to show that in order to solve these problems we can use probabilistic methods to demonstrate the existence of the graphs without actually constructing them. (It should be noted that we never use more than the convenient language of probability theory, since all the probabilistic arguments we need can be replaced by counting the number of objects in various sets.) This phenomenon is not confined to graph theory and combinatorics; in the last decade or two probabilistic methods have been used with striking success in Fourier analysis, in the theory of function spaces, in number theory, in the geometry of Banach spaces, etc. However, there is no area where probabilistic (or counting) methods are more natural than in combinatorics.