Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$
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| Format: | Preprint |
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2026
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| _version_ | 1866915898976108544 |
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| author | Adak, Rajat Bakshi, Yash Chandran, L. Sunil Nanoti, Saraswati Girish |
| author_facet | Adak, Rajat Bakshi, Yash Chandran, L. Sunil Nanoti, Saraswati Girish |
| contents | 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 distinct linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and asks for the least integer $N$ such that every red--blue coloring of $\{1, 2, \dots, N\}$ must yield 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 off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$ Adak, Rajat Bakshi, Yash Chandran, L. Sunil Nanoti, Saraswati Girish Combinatorics 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 distinct linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and asks for the least integer $N$ such that every red--blue coloring of $\{1, 2, \dots, N\}$ must yield 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 off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations. |
| title | Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.28216 |