On Some Properties of Accessible Sets
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909654862266368 |
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| author | Quester, Oscar |
| author_facet | Quester, Oscar |
| contents | A set $D \subseteq \mathbb{N}$ is called $r$-large if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic arithmetic progressions $a,a+d,...,a+(k-1)d$ with gap $d \in D$. Closely related to largeness is accessibility; a set $D \subseteq \mathbb{N}$ is called $r$-accessible if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic sequences $x_1,x_2,...,x_k$ with $x_{i+1}-x_{i} \in D$. It is known that if $D \subseteq \mathbb{N}$ is $2$-large, then the gaps between elements in $D$ cannot grow exponentially. In this paper, we show that if $D$ is $2$-accessible, then the gaps between elements in $D$ cannot grow much faster than exponentially. Additionally, we show that the notion of accessibility is equivalent to that of topological recurrence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05356 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Some Properties of Accessible Sets Quester, Oscar Combinatorics A set $D \subseteq \mathbb{N}$ is called $r$-large if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic arithmetic progressions $a,a+d,...,a+(k-1)d$ with gap $d \in D$. Closely related to largeness is accessibility; a set $D \subseteq \mathbb{N}$ is called $r$-accessible if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic sequences $x_1,x_2,...,x_k$ with $x_{i+1}-x_{i} \in D$. It is known that if $D \subseteq \mathbb{N}$ is $2$-large, then the gaps between elements in $D$ cannot grow exponentially. In this paper, we show that if $D$ is $2$-accessible, then the gaps between elements in $D$ cannot grow much faster than exponentially. Additionally, we show that the notion of accessibility is equivalent to that of topological recurrence. |
| title | On Some Properties of Accessible Sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.05356 |