Combinatorics Seminar

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 $\{x,x+y,xy\}$.