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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.03222 |
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| _version_ | 1866910829678428160 |
|---|---|
| author | Wang, Runze |
| author_facet | Wang, Runze |
| contents | Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic method, we prove that
\begin{align*}
\frac{|H_G(S)|-1}{|H_G(S)|}|G|+\Biggl\lceil\biggl(\frac{|G|}{|H_G(S)|}\biggr)^{1-|H_G(S)|/|S|}\Biggr\rceil\le N_{G,\ S}\le \biggl\lfloor\frac{|S|-1}{|S|}|G|\biggr\rfloor+1,
\end{align*}
where $H_G(S)$ is the stabilizer of $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the size of sets avoiding a general structure Wang, Runze Combinatorics Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic method, we prove that \begin{align*} \frac{|H_G(S)|-1}{|H_G(S)|}|G|+\Biggl\lceil\biggl(\frac{|G|}{|H_G(S)|}\biggr)^{1-|H_G(S)|/|S|}\Biggr\rceil\le N_{G,\ S}\le \biggl\lfloor\frac{|S|-1}{|S|}|G|\biggr\rfloor+1, \end{align*} where $H_G(S)$ is the stabilizer of $S$. |
| title | On the size of sets avoiding a general structure |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.03222 |