Joel Moreira

Partition Regularity of Pythagorean Pairs

An influential and still open question of Erdos and Graham in Ramsey theory asks whether any finite colouring of the natural numbers yields a monochromatic solution to the equation x^2+y^2=z^2. In recent joint work with Nikos Frantzikinakis and Oleksiy Klurman we answer a simpler question and show that in any finite colouring of the natural numbers there exists a solution to x^2+y^2=z^2 with two of the variables in the same colour. The proof combines techniques from ergodic theory and number theory. In the talk I will explain how ergodic theory can be used in this kind of combinatorial problems, and I will outline the main dynamical steps in the proof.