Conference Global Portuguese Mathematicians 

14/07/2017, 11:00 — 12:00 — Amphitheatre Pa1, Mathematics Building
Joel Moreira, Northwestern University

Monochromatic sums and products

Is it possible to color the natural numbers with finitely many colors, so that whenever $x$ and $y$ are of the same color, their sum $x+y$ has a different color? A 1916 theorem of I. Schur tells us that the answer is *no*. In other words, for any finite coloring of $\mathbb{N}$, there exist $x$ and $y$ such that the triple $\{x,y,x+y\}$ is monochromatic (i.e. all terms have the same color). A similar result holds if one replaces the sum $x+y$ with the product $xy$, however, it is still unknown whether one can finitely color the natural numbers in a way that no quadruple $\{x,y,x+y,xy\}$ is monochromatic!

A recent partial solution to this problem states that any finite coloring of the natural numbers yields a monochromatic triple $\{x,x+y,xy\}$. In order to present the main ideas underlying this proof, I will review some classical results in Ramsey theory and explain how ergodic theory and dynamical systems can be used to answer questions in that field.