The Sarnak conjecture asks whether there is a non-trivial correlation between the Mobius function and any sequence coming from a topological system with zero entropy. This conjecture was widely studied in the last decade, and it had many connections to problems in combinatorics and number theory. Nowadays many partial answers to this conjecture are known. In particular, it is known that the conjecture holds for nilsequences.

The partition regularity problem studies problems of the following form. Given an equation of multi variables, for any finite coloring of integers, is there always a monochromatic solution to the equation? This question was well understood when the equation is linear, but the case for quadratic equations is widely open. In this talk, I will explain how the study of the Sarnak’s Conjecture can help with the study of the partition regularity problem.