On Rado's single equation theorem

Abstract

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

21pp