Rainbow and Gallai-Rado numbers involving binary function equations

Abstract

Let $\mathcal{E}$, ${\mathcal{E}}_1$, and ${\mathcal{E}}_2$ be equations, $n$ and $k$ be positive integers. The rainbow number $\mathrm{rb}([n],\mathcal{E})$ is difined as the minimum number of colors such that for every exact $(\mathrm{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a rainbow solution of $\mathcal{E}$. The Gallai-Rado number ${\mathrm{GR}}_k({\mathcal{E}}_1:{\mathcal{E}}_2)$ is defined as the minimum positive integer $N$, if it exists, such that for all $n\ge N$, every $k$-colored $[n]$ contains either a rainbow solution of ${\mathcal{E}}_1$ or a monochromatic solution of ${\mathcal{E}}_2$. In this paper, we get some exact values of rainbow and Gallai-Rado numbers involving binary function equations. We also provide an algorithm to calculate the rainbow numbers of nonlinear binary function equations.