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

Abstract

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 linear equations $({\mathcal{E}}_1,{\mathcal{E}}_2)$ and seeks the least integer $N$ such that every red--blue coloring of $\{1,2,\dots ,N\}$ contains 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 two-color off-diagonal Rado numbers for the nonhomogeneous linear equations $x+y+c=z$ and $x+y+k=z$, where $c\le k$. In the discrete setting, $c$ and $k$ are nonnegative integers, whereas in the continuous setting, they are nonnegative real numbers. We determine the exact discrete and continuous two-color off-diagonal Rado numbers for this pair of shifted Schur equations.

12 pages; revised exposition with improved presentation and English, and removal of redundant steps. Main results remain unchanged