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

TL;DR

This paper determines the exact off-diagonal Rado number for the system of equations $x+y+c=z$ and $x+qy=z$, extending Ramsey theory for linear equations to non-homogeneous cases.

Contribution

It provides the first exact values of off-diagonal Rado numbers for non-homogeneous linear equations of specific forms.

Findings

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

Results extend Ramsey theory to non-homogeneous equations.

Provides a foundation for further research in off-diagonal Rado numbers.

Abstract

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations $(\mathcal{E}_1,\mathcal{E}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,...,N\}$ forces either a red solution of the equation $\mathcal{E}_1$ or a blue solution of the equation $\mathcal{E}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.