 
Summary: Algebraic and probabilistic methods in Discrete Mathematics
Noga Alon
Abstract
Combinatorics is an essential component of many mathematical areas, and its study has ex
prienced an impressive growth in recent years. This survey contains a discussion of two of the
main general techniques that played a crucial role in the development of modern combinatorics;
algebraic methods and probabilistic methods. Both techniques are illustrated by examples, where
the emphasis is on the basic ideas and the connection to other areas.
1 Introduction
Mathematical Research deals with ideas that can be meaningful to everybody and there is no doubt
that it also lies behind most of the major advances in Science and Technology. Yet, mathematicians
often tend to formulate their questions, results and thoughts in a way that is comprehensible only
to their colleagues that work in a closely related area. One of the goals of the conference "Visions in
Mathematics" was to try and present the main areas in mathematics in a way that can be interesting
to a general mathematical audience, and possibly even to a general scientific audience. Although
this is a difficult task, it is not impossible, and I believe that many of the lectures achieved this goal.
Following the spirit of the conference, this survey is also aimed to a general mathematical audi
ence. I try to explain two of the main techniques that played a crucial role in the development of
modern combinatorics: algebraic techniques and probabilistic methods. The focus is on basic ideas,
rather than on technical details, and the techniques are illustrated by examples that demonstrate
