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arXiv · 2602.23954

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

Rajat Adak·Yash Bakshi·L. Sunil Chandran·Saraswati Girish Nanoti
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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.

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Rajat Adak, Yash Bakshi, L. Sunil Chandran, Saraswati Girish Nanoti. 2026-02-27. Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$. https://arxiv.org/abs/2602.23954

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