Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.19650 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913336644337664 |
|---|---|
| author | Tao, T. Y. Yang, Neil N. Y. |
| author_facet | Tao, T. Y. Yang, Neil N. Y. |
| contents | Hindman conjectured that any finite partition of $\mathbb{N}$ has a monochromatic $\{x,y,x+y,xy\}$. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen's result to any semiring $(S,+,\cdot)$ such that $Ss$ is piecewise syndetic for all $s\in S$. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock's IP$_r^*$ Szemerédi Theorem, and discussed the case when the operation is not commutative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finding Product and Sum Patterns in non-commutative settings Tao, T. Y. Yang, Neil N. Y. Combinatorics Hindman conjectured that any finite partition of $\mathbb{N}$ has a monochromatic $\{x,y,x+y,xy\}$. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen's result to any semiring $(S,+,\cdot)$ such that $Ss$ is piecewise syndetic for all $s\in S$. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock's IP$_r^*$ Szemerédi Theorem, and discussed the case when the operation is not commutative. |
| title | Finding Product and Sum Patterns in non-commutative settings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.19650 |