On a problem of Brown, Erdős and Sós
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910402978250752 |
|---|---|
| author | Letzter, Shoham Sgueglia, Amedeo |
| author_facet | Letzter, Shoham Sgueglia, Amedeo |
| contents | Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph not containing a subhypergraph with $k$ edges on at most $s$ vertices. Recently, Delcourt and Postle, building on work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that the limit $\lim_{n \to \infty} n^{-2} f^{(3)}(n;k+2,k)$ exists for all $k \ge 2$, solving an old problem of Brown, Erdős and Sós (1973). Meanwhile, Shangguan and Tamo asked the more general question of determining if the limit $\lim_{n \to \infty} n^{-t} f^{(r)}(n;k(r-t)+t,k)$ exists for all $r>t\ge 2$ and $k \ge 2$.
Here we make progress on their question. For every even $k$, we determine the value of the limit when $r$ is sufficiently large with respect to $k$ and $t$. Moreover, we show that the limit exists for $k \in \{5,7\}$ and all $r > t \ge 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03856 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a problem of Brown, Erdős and Sós Letzter, Shoham Sgueglia, Amedeo Combinatorics Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph not containing a subhypergraph with $k$ edges on at most $s$ vertices. Recently, Delcourt and Postle, building on work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that the limit $\lim_{n \to \infty} n^{-2} f^{(3)}(n;k+2,k)$ exists for all $k \ge 2$, solving an old problem of Brown, Erdős and Sós (1973). Meanwhile, Shangguan and Tamo asked the more general question of determining if the limit $\lim_{n \to \infty} n^{-t} f^{(r)}(n;k(r-t)+t,k)$ exists for all $r>t\ge 2$ and $k \ge 2$. Here we make progress on their question. For every even $k$, we determine the value of the limit when $r$ is sufficiently large with respect to $k$ and $t$. Moreover, we show that the limit exists for $k \in \{5,7\}$ and all $r > t \ge 2$. |
| title | On a problem of Brown, Erdős and Sós |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.03856 |