Algorithms, Combinatorics and Optimization Seminar Joel Moreira
Northwestern University
Title: Monochromatic configurations in finite colorings of $\mathbb{N}$

Abstract: 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. has 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! In this talk I present a recent partial solution to this problem, showing that any finite coloring of the natural numbers yields a monochromatic triple {x,x+y,xy}.

Date: Thursday, October 20, 2016
Time: 3:30 pm
Location: Wean Hall 8220