The structure of Sidon set systems
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914684018360320 |
|---|---|
| author | Wötzel, Maximilian |
| author_facet | Wötzel, Maximilian |
| contents | A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$ of a Sidon system consisting of $k$-subsets of the first $n$ positive integers satisfies $C_k n^{k-1}\leq F_k(n) \leq \binom{n-1}{k-1}+n-k$ for some constant $C_k$ only depending on $k$. We close the gap by proving an essentially tight structural result that in particular implies $F_k(n)\geq (1-o(1))\binom{n}{k-1}$. We also use this to establish a result about the size of the largest Sidon system in the binomial random family $\binom{[n]}{k}_p$. Extensions to $h$-fold sumsets for any fixed $h\geq 3$ are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14011 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The structure of Sidon set systems Wötzel, Maximilian Combinatorics Number Theory 11B13 (Primary) 05B10 (Secondary) A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$ of a Sidon system consisting of $k$-subsets of the first $n$ positive integers satisfies $C_k n^{k-1}\leq F_k(n) \leq \binom{n-1}{k-1}+n-k$ for some constant $C_k$ only depending on $k$. We close the gap by proving an essentially tight structural result that in particular implies $F_k(n)\geq (1-o(1))\binom{n}{k-1}$. We also use this to establish a result about the size of the largest Sidon system in the binomial random family $\binom{[n]}{k}_p$. Extensions to $h$-fold sumsets for any fixed $h\geq 3$ are also obtained. |
| title | The structure of Sidon set systems |
| topic | Combinatorics Number Theory 11B13 (Primary) 05B10 (Secondary) |
| url | https://arxiv.org/abs/2211.14011 |