Introduction to the probabilistic method

The probabilistic method is an efficient technique to prove the existence of combinatorial objects having some specific properties. It is based on the probability theory but it can be used to prove theorems which have nothing to do with probability. The probabilistic method is now widely used and considered as a basic knowledge. It is now exposed in many introductory books to graph theory like [8]. In these notes, we give the most common tools and techniques of this method. The interested reader who would like to know more complicated techniques is referred to the books of Alon and Spencer [5] and Molloy and Reed [21]. We assume that the reader is familiar with basic graph theory. Several books (see e.g. [8, 11, 29]) give nice introduction to graph theory.