 
Summary: Probabilistic Proofs of Existence of Rare Events
Noga Alon
Department of Mathematics
Sackler Faculty of Exact Sciences
Tel Aviv University
RamatAviv, 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
