On Rado's single equation theorem

TL;DR

This paper proves a bound on the size of a set needed to guarantee monochromatic solutions to a specific linear equation under any coloring, extending Rado's theorem.

Contribution

It establishes an explicit exponential bound for the smallest N ensuring monochromatic solutions in any r-coloring, refining previous results on Rado's single equation theorem.

Findings

Existence of N < exp(r^{2+o(1)}) for monochromatic solutions

Monochromatic solutions occur for all non-zero integers a, b

The bound improves understanding of colorings in additive combinatorics

Abstract

We show that for non-zero integers $a$ and $b$ there is a natural number $N < \exp(r^{2+o_{a,b;r\rightarrow \infty}(1)})$ such that in any $r$-colouring of $\{1,\dots,N\}$ there are $x,y,z$, all in the same colour class, such that $ax-ay=bz$.