Gallai-Schur Triples and Related Problems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917940532609024 |
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| author | Mao, Yaping Robertson, Aaron Wang, Jian Yang, Chenxu Yang, Gang |
| author_facet | Mao, Yaping Robertson, Aaron Wang, Jian Yang, Chenxu Yang, Gang |
| contents | Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer $GS(r)$ such that every $r$-coloring of $\{1,2,\dots,GS(r)\}$ admits either a rainbow or monochromatic solution to $x+y=z$. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when $x \neq y$, we consider $x+y+b=z$ and $x+y<z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y<z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gallai-Schur Triples and Related Problems Mao, Yaping Robertson, Aaron Wang, Jian Yang, Chenxu Yang, Gang Combinatorics 05D10 Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer $GS(r)$ such that every $r$-coloring of $\{1,2,\dots,GS(r)\}$ admits either a rainbow or monochromatic solution to $x+y=z$. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when $x \neq y$, we consider $x+y+b=z$ and $x+y<z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y<z$. |
| title | Gallai-Schur Triples and Related Problems |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2502.21221 |