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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 colorssuch 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 minimumpositive 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 amonochromatic solution of $\mathcal{E}_2$. In this paper, we get some exactvalues of rainbow and Gallai-Rado numbers involving binary function equations.We also provide an algorithm to calculate the rainbow numbers of nonlinearbinary function equations.

Rainbow and Gallai-Rado numbers involving binary function equations