Extremal combinatorics deals with the class of problems that ask to determine the maximum, or minimum, possible size of a collection of finite objects with certain properties. This is a central branch of combinatorics that has seen impressive growth in recent years, many of whose problems arise naturally in various fields in mathematics and elsewhere.

In this talk I will discuss classical results, recent developments, and interesting open problems in the area of extremal combinatorics. In particular, I will focus on Ramsey theory, whose underlying philosophy can be described as showing that complete disorder is impossible'. I will then move on to extremal graph theory and extremal set theory, where the objects of study are graphs and set systems, respectively.

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