Probabilistic Methods in Combinatorics

In 1947 Paul Erdös [8] began what is now called the probabilistic method. He showed that if (fc)2 ' ' < i then there exists a graph G on n vertices with clique number u)(G) < k and independence number ct(G) < k. (In terms of the Ramsey function, R(k,k) > n.) In modern language he considered the random graph G(n, .5) as described below. For each fc-set S let Bs denote the "bad" event that S is either a clique or an independent set. Then Pr [i?s] = 2 " W so that ^Pr [£ , s ] < 1, hence AB s ^ 0 and a graph satisfying AB s must exist. In 1961 Erdös with Alfred Rényi [11] began the systematic study of random graphs. Formally G(n,p) is a probability space whose points are graphs on a fixed labelled set of n vertices and where every pair of vertices is adjacent with independent probability p. A graph theoretic property A becomes an event. Whereas in the probabilistic method one generally requires only Pr[A] > 0 from which one deduces the existence of the desired object, in random graphs the estimate of Pr[A] is the object itself. Let A denote connectedness. In their most celebrated result Erdös and Rényi showed that if p = p(n) = ^ + ^ then Pr[A] -> exp(-e~ ) . We give [2], [6] as general references for these topics. Although pure probability underlies these fields, most of the basic results use fairly straightforward methods. The past ten years (our emphasis here) have seen the use of a number of more sophisticated probability results. The Chernoff bounds have been enhanced by inequalities of Janson and Talagrand and new appreciation of an inequality of Azuma. Entropy is used in new ways. In its early days the probabilistic method had a magical quality — where is the graph that Erdös in 1947 proved existed? With the rise of theoretical computer science these questions take on an algorithmic tone — having proven the existence of a graph or other structure, can it be constructed in polynomial time? A recent success of Beck allows the Lovâsz local lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asymptopia by David Aldous. There the asymptotic behavior of random objects is given by an infinite object, allowing powerful noncombinatorial tools to be used.