Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$

The study of Ramsey-type problems for linear equations originated with Schur’s theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of distinct linear equations $({\mathcal{E}}_1,{\mathcal{E}}_2)$ and asks for the least integer $N$ such that every red–blue coloring of $\{1,2,\dots ,N\}$ must yield either a red solution to ${\mathcal{E}}_1$ or a blue solution to ${\mathcal{E}}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $({\mathcal{E}}_1,{\mathcal{E}}_2)$. In this work, we study the discrete and continuous off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations.