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arXiv · 2603.28216

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

Rajat Adak·Yash Bakshi·L. Sunil Chandran·Saraswati Girish Nanoti
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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 linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and seeks the least integer $N$ such that every red--blue coloring of $\{1,2,\dots,N\}$ contains 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 two-color off-diagonal Rado numbers for the nonhomogeneous linear equations $x+y+c=z$ and $x+y+k=z$, where $c\leq k$. In the discrete setting, $c$ and $k$ are nonnegative integers, whereas in the continuous setting, they are nonnegative real numbers. We determine the exact discrete and continuous two-color off-diagonal Rado numbers for this pair of shifted Schur equations.

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Rajat Adak, Yash Bakshi, L. Sunil Chandran, Saraswati Girish Nanoti. 2026-08-11. Off-diagonal Rado numbers for $x+y+c=z$ and $x+y+k=z$. https://arxiv.org/abs/2603.28216

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