Partition regularity of generalised Fermat equations

Abstract: Let $\alpha,\beta,\gamma\in\mathbb{N}$. We prove that given an $r$-colouring of $\mathbb{F}p$ with $p$ prime, there are more than $c{r,\alpha,\beta,\gamma} p^2$ solutions to the equation $x^\alpha+y^\beta=z^\gamma$ with all of $x,y,z$ of the same colour. Here $c_{r,\alpha,\beta,\gamma}>0$ is some constant depending on the number of colours and the exponents in the equation. This is already a new result for $\alpha=\beta=1$ and $\gamma=2$, that is to say for the equation $x+y=z^2$.