On the quadratic 8-edge case of the Brown-Erdős-Sós problem
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912650587275264 |
|---|---|
| author | Pikhurko, Oleg Sun, Shumin |
| author_facet | Pikhurko, Oleg Sun, Shumin |
| contents | Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erdős and Sós conjectured in 1973 that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. Recently, Delcourt and Postle settled the conjecture and their approach was generalised by Shangguan to every uniformity $r\ge 4$: the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$.
The value of the limit is currently known for $k\in \{2,3,4,5,6,7\}$ due to various results authored by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. In this paper we consider the case $k=8$, determining the value of the limit for each $r\ge 4$ and presenting a lower bound for $k=3$ that we conjecture to be sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the quadratic 8-edge case of the Brown-Erdős-Sós problem Pikhurko, Oleg Sun, Shumin Combinatorics Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erdős and Sós conjectured in 1973 that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. Recently, Delcourt and Postle settled the conjecture and their approach was generalised by Shangguan to every uniformity $r\ge 4$: the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$. The value of the limit is currently known for $k\in \{2,3,4,5,6,7\}$ due to various results authored by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. In this paper we consider the case $k=8$, determining the value of the limit for each $r\ge 4$ and presenting a lower bound for $k=3$ that we conjecture to be sharp. |
| title | On the quadratic 8-edge case of the Brown-Erdős-Sós problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.01739 |