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

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.

💡 Research Summary

The paper studies a two‑color off‑diagonal Rado number for a pair of non‑homogeneous linear equations: E₁: x + y + c = z (assigned to the red colour) and E₂: x + q y = z (assigned to the blue colour). For positive integers c and q the authors define R₂(c,q) as the smallest N such that every red‑blue colouring of the set {1,…,N} forces either a red monochromatic solution of E₁ or a blue monochromatic solution of E₂. The main contribution is an exact formula for R₂(c,q) covering all possible parity combinations of c and q.

The paper begins with a concise review of classical Rado theory, Schur’s theorem, and earlier work on diagonal and off‑diagonal Rado numbers. It points out that while diagonal cases (the same equation in both colours) have been extensively studied, the off‑diagonal situation for non‑homogeneous equations remains largely open.

The authors present three mutually exclusive cases:

1. Both c and q are odd. In this situation they construct a simple parity‑based colouring (even numbers red, odd numbers blue) that avoids a red solution to E₁ and a blue solution to E₂ simultaneously. Consequently no finite N can guarantee a monochromatic solution, and therefore R₂(c,q)=∞.

2. q = 1 and c is even. They first give a lower bound by colouring the interval