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 $\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum number of colors such that for every exact $(\operatorname{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a rainbow solution of $\mathcal{E}$. The Gallai-Rado number $\operatorname{GR}_k(\mathcal{E}_1:\mathcal{E}_2)$ 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.