Monochromatic sums and products

TL;DR

This paper proves that in any r-colouring of a finite field, one colour class contains a significant number of quadruples involving sums and products, highlighting inherent combinatorial structures.

Contribution

It establishes a lower bound on the number of specific sum-product quadruples within a single colour class in finite fields.

Findings

Existence of large monochromatic sum-product quadruples

Quantitative bounds on quadruple counts in finite fields

Structural insights into colourings of finite fields

Abstract

Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.