Ergodic theory at the service of combinatorics and number theory, Vitaly Bergelson,
8/7/2023, Warsaw - Numbers in the Universe school and conference.

Lecture 1. Ramsey Theory and Furstenberg’s correspondence principle.
In the first half of this talk we will formulate and discuss various results from combinatorics and additive number theory which illustrate the main principles of Ramsey theory. In the second half, we will discuss Furstenberg’s ergodic approach to Szemerédi’s theorem on arithmetic progressions and formulate Furstenberg’s correspondence principle which allows one to derive various combinatorial results from ergodic-theoretical multiple recurrence theorems. We will conclude with the discussion of various extensions and refinements of Szemerédi’s theorem.