Ramsey Theory

Abstract

Ramsey theory is one of the central topics in modern combinatorics, with applications and relations to many other parts of mathematics. This course introduces many of the fundamental results, techniques, and problems in Ramsey theory, covering both classical results and very recent breakthroughs. Applications to computer science, geometry, number theory, and other fields will also be discussed.

Objective

Students will learn many of the main techniques used in the study of Ramsey theory. They will be expected to understand these techniques, the results they prove, and the significance of these results in other areas of mathematics. We will also expect them to be able to apply these techniques to related problems.

Content

Ramsey theory refers to a wide body of results and ideas, which can be summarized as "complete disorder is impossible". Indeed, Ramsey-theoretic results imply that, in a very wide range of disparate settings, any sufficiently large system contains a large structured subsystem. The origins of Ramsey theory can be traced back more than a century, and it has been studied as a central part of combinatorics since at least the 1940s. Yet despite decades of intense research and tremendous achievements, there remain fundamental questions which are still wide open. Many of these questions have seen remarkable progress in recent years, including a number of spectacular breakthroughs in 2023. The goal of this class is to introduce students to this beautiful and complex subject, with an equal focus on classical results, recent breakthroughs, widely-applicable techniques, and applications to other fields such as computer science, geometry, and number theory.

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