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

2026-02-27

AI Summary

This study investigates the two-color non-diagonal Rado number for the system of inhomogeneous linear equations $x + y + c = z$ and $x + qy = z$, defined as the smallest integer $N$ such that any red–blue coloring of $\{1, 2, \dots, N\}$ yields a monochromatic solution to at least one of the equations. By integrating techniques from Ramsey theory, extremal combinatorics, explicit counterexample constructions, and inductive arguments, the authors fully determine the exact value of this Rado number, denoted $R_2(c,q)$. The result not only establishes a precise threshold but also extends the scope of Rado’s theorem to inhomogeneous and asymmetric settings, offering a new paradigm of exact solutions in Ramsey-type problems for linear equations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Innovation

• off-diagonal Rado number
• non-homogeneous linear equations
• Ramsey theory
• Schur-type problems
• two-color coloring