Seminar 10.3.13

On {x + y, xy} patterns in large sets of countable fields


Oct 3 2013 – 3:00pm – 4:00 pm




Joel Moreira (Ohio State)


A theorem of I. Schur, proved in 1916, states that for every finite coloring of the positive integer numbers there are positive integers x and y such that the set {x,y,x+y} is monochromatic. One can easily deduce that there are also positive integers x and y such that the set {x,y,xy} is monochromatic. This leads to the following question: 
“Is it true that for a finite coloring of the positive integer numbers there exist x and y such that {x,y,x+y,xy} is monochromatic?”
This talk will describe recent progress, in joint work with V. Bergelson, on an analog of this problem with the set of positive integers replaced with a countably infinite field. We introduce the notion of double Følner sequence in a field K. This gives rise to an upper density on K that is invariant under both addition and multiplication. Next we show that a set B⊂K with positive upper density contains a configuration of the form {x+y,xy}. We also show that for a finite coloring of K there exist x and y in K such that {x,x+y,xy} is monochromatic.