Rainbow Free Colorings and Rainbow Numbers for $x-y=z^2$

Abstract

An exact r-coloring of a set $S$ is a surjective function $c:S\to \{1,2,\dots ,r\}$. A rainbow solution to an equation over $S$ is a solution such that all components are a different color. We prove that every 3-coloring of $\mathbb{N}$ with an upper density greater than $(4^s-1)/(3\cdot 4^s)$ contains a rainbow solution to $x-y=z^k$. The rainbow number for an equation in the set $S$ is the smallest integer $r$ such that every exact $r$-coloring has a rainbow solution. We compute the rainbow numbers of ${\mathbb{Z}}_p$ for the equation $x-y=z^k$, where $p$ is prime and $k\ge 2$.