Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866918149290459136 |
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| author | Csajbók, Bence Nagy, Zoltán Lóránt |
| author_facet | Csajbók, Bence Nagy, Zoltán Lóránt |
| contents | A subset $S$ of an abelian group $G$ is called $3$-$\mathrm{AP}$ free if it does not contain a three term arithmetic progression. Moreover, $S$ is called complete $3$-$\mathrm{AP}$ free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a $3$-$\mathrm{AP}$ free set, which is necessarily complete. In this paper we are interested in the minimum size of complete $3$-$\mathrm{AP}$ free sets. We define and study saturation w.r.t. $3$-$\mathrm{AP}$s and present constructions of small complete $3$-$\mathrm{AP}$ free sets and $3$-$\mathrm{AP}$ saturating sets for several families of vector spaces and cyclic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups Csajbók, Bence Nagy, Zoltán Lóránt Combinatorics Number Theory 51E21, 11B25, 05D05 A subset $S$ of an abelian group $G$ is called $3$-$\mathrm{AP}$ free if it does not contain a three term arithmetic progression. Moreover, $S$ is called complete $3$-$\mathrm{AP}$ free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a $3$-$\mathrm{AP}$ free set, which is necessarily complete. In this paper we are interested in the minimum size of complete $3$-$\mathrm{AP}$ free sets. We define and study saturation w.r.t. $3$-$\mathrm{AP}$s and present constructions of small complete $3$-$\mathrm{AP}$ free sets and $3$-$\mathrm{AP}$ saturating sets for several families of vector spaces and cyclic groups. |
| title | Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups |
| topic | Combinatorics Number Theory 51E21, 11B25, 05D05 |
| url | https://arxiv.org/abs/2401.06283 |