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Rainbow Free Colorings and Rainbow Numbers for $x-y=z^2$ (2305.15133v1)

Published 24 May 2023 in math.CO and math.NT

Abstract: An exact r-coloring of a set $S$ is a surjective function $c:S \rightarrow {1, 2, \ldots,r}$. 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 $(4s-1)/(3 \cdot 4s)$ contains a rainbow solution to $x-y=zk$. 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 $x-y=zk$, where $p$ is prime and $k\geq 2$.

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