Monochromatic sums and products

Monochromatic sums and products, Discrete Analysis 2016:5, 48pp.

An old and still unsolved problem in Ramsey theory asks whether if the positive integers are coloured with finitely many colours, then there are positive integers $x$ and $y$ such that $x, y, x+y$ and $xy$ all have the same colour. In fact, it is not even known whether it is always possible to find $x$ and $y$ such that $x+y$ and $xy$ have the same colour.

This paper is about the corresponding question when $\mathbb{N}$ is replaced by a finite field $\mathbb{F}_p$, and gives a positive answer. More than that, it proves that a positive fraction of the quadruples $(x,y,x+y,xy)$ are monochromatic.

The result is interesting for several reasons. One is that the standard tools of Ramsey theory appear to be hopelessly inadequate when they are applied to questions that mix addition and multiplication, so the fact that the authors have obtained a positive result of this kind is surprising and may well have further ramifications. Another is that several people have tried, without much success, to apply techniques from additive combinatorics to colouring problems. The techniques work well for many density problems, from which one can of course deduce colouring results. Until now the challenge has been to get them to work for colouring problems when the corresponding density statements are false, as is the case here (since the set of numbers between $p/3$ and $2p/3$ does not even contain a triple of the form $(x,y,x+y)$).

A third reason, which is almost implied by the previous two, is that the paper introduces some striking new techniques. One of these techniques is to “smooth” the colouring in a way that converts the count of quadruples $(x,y,x+y,xy)$ into a count of purely linear configurations, thereby making the problem more amenable to conventional Ramsey-theoretic techniques. It also uses deep character sum estimates from number theory.

For all these reasons, the paper will repay careful study by those who work in additive combinatorics.