Van der Waerden type theorem for amenable groups and FC-groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912131805347840 |
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| author | Parini, Emilio |
| author_facet | Parini, Emilio |
| contents | We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is monochromatic} \}$, is left IP$^{\ast}$. This partially solves a conjecture of V. Bergelson and R. McCutcheon. Moreover, we prove that the result holds for $G^m$ if $G$ is an FC-group, i.e., all conjugacy classes of $G$ are finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Van der Waerden type theorem for amenable groups and FC-groups Parini, Emilio Group Theory Combinatorics We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is monochromatic} \}$, is left IP$^{\ast}$. This partially solves a conjecture of V. Bergelson and R. McCutcheon. Moreover, we prove that the result holds for $G^m$ if $G$ is an FC-group, i.e., all conjugacy classes of $G$ are finite. |
| title | Van der Waerden type theorem for amenable groups and FC-groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2411.15987 |