Oct. 20, 3:30pm, Wean 8220

Joel Moreira, Northwestern University

Monochromatic configurations in finite colorings of ℕ

Joel Moreira, Northwestern University

Monochromatic configurations in finite colorings of 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 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*}.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.