The Probabilistic Method, Random Graphs and Stein’s Method

The Probabilistic Method has been initiated by Paul Erdős [6] in order to prove the existence of certain combinatorial objects. The principle idea is to define a proper probability distribution on a class of (discrete) objects and to show that the probability of a certain property is positive. Of course this also proves that there exists such an object with this property. We will apply this approach to various problems on random graphs. However, the main goal of this course is to give an introduction to Stein’s method that proves asymptotic normality for sums of (in some sense) weakly dependent random variables. This method has turned out to be very successful, in particular in random graph problems. There is vast literature on these topics. We just mention few books that are exclusively devoted to them [1, 5, 9, 10].