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

TL;DR

This paper determines the exact off-diagonal Rado number for the linear systems $x+y+c=z$ and $x+y+k=z$ in both discrete and continuous two-color settings, advancing understanding of Ramsey-type problems for these equations.

Contribution

It provides the first exact values of the off-diagonal Rado number for the specific non-homogeneous systems $x+y+c=z$ and $x+y+k=z$ in both discrete and continuous cases.

Findings

Exact two-color discrete off-diagonal Rado number $R_2(c,k)$ determined.

Exact continuous off-diagonal Rado number $R_2(c,k)$ determined.

Results extend the understanding of Ramsey problems for non-homogeneous linear systems.

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 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.