Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917299655540736 |
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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 | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23954 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$ Adak, Rajat Bakshi, Yash Chandran, L. Sunil Nanoti, Saraswati Girish Combinatorics Discrete Mathematics 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. |
| title | Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$ |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2602.23954 |