On the two-colour Rado number for $\sum_{i=1}^m a_ix_i=c$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912079771860992 |
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| author | Arora, Ishan Dwivedi, Srashti Tripathi, Amitabha |
| author_facet | Arora, Ishan Dwivedi, Srashti Tripathi, Amitabha |
| contents | Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the interval $[1,N]$ admits a monochromatic solution to the given equation. We introduce the concept of $t$-distributability of sets of positive integers, and determine exact values whenever possible, and upper and lower bounds otherwise, for the Rado numbers when the set $\{a_1,\ldots,a_{m-1}\}$ is $2$-distributable or $3$-distributable, $a_m=-1$, and $r=2$. This generalizes previous works by several authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the two-colour Rado number for $\sum_{i=1}^m a_ix_i=c$ Arora, Ishan Dwivedi, Srashti Tripathi, Amitabha Combinatorics 05C55 (Primary), 05D10 (Secondary) Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the interval $[1,N]$ admits a monochromatic solution to the given equation. We introduce the concept of $t$-distributability of sets of positive integers, and determine exact values whenever possible, and upper and lower bounds otherwise, for the Rado numbers when the set $\{a_1,\ldots,a_{m-1}\}$ is $2$-distributable or $3$-distributable, $a_m=-1$, and $r=2$. This generalizes previous works by several authors. |
| title | On the two-colour Rado number for $\sum_{i=1}^m a_ix_i=c$ |
| topic | Combinatorics 05C55 (Primary), 05D10 (Secondary) |
| url | https://arxiv.org/abs/2410.16051 |