Probabilistic Proofs of Existence of Rare Events Department of Mathematics Summary: Probabilistic Proofs of Existence of Rare Events Noga Alon Department of Mathematics Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Tel Aviv 69978 ISRAEL 1. The Local Lemma In a typical probabilistic proof of a combinatorial result, one usually has to show that the probability of a certain event is positive. However, many of these proofs actually give more and show that the probability of the event considered is not only positive but is large. In fact, most probabilistic proofs deal with events that hold with high probability, i.e., a probability that tends to 1 as the dimensions of the problem grow. For example, recall that a tournament on a set V of n players is a set of ordered pairs of distinct elements of V , such that for every two distinct elements x and y of V , either (x, y) or (y, x) is in the tournament, but not both. The name tournament is natural, since one can think on the set V as a set of players in which each pair participates in a single match, where (x, y) is in the tournament iff x defeated y. As shown by Erd¨os in [Er] for each k 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that a Collections: Mathematics