On the quadratic 8-edge case of the Brown-Erdős-Sós problem
Oleg Pikhurko, Shumin Sun
TL;DR
This work determines the quadratic limit for the Brown–Erdős–Sós problem at $k=8$ for all uniformities $r\ge4$, proving $\pi(r,8)=\frac{1}{r^2-r}$ and supplying a nontrivial lower bound $\pi(3,8)\ge\frac{3}{16}$ (conjectured sharp). The authors extend the Delcourt–Postle and Shangguan framework via a two-tier merging/weight approach: first decompose $r$-graphs into 1- and 2-clusters, then assign carefully crafted weights to control edge counts, ensuring a global $|G|$ bound of $\frac{1}{\binom{r}{2}}\binom{n}{2}$. A key technical element is a structural lemma that classifies large 2-clusters, together with a trimming lemma that preserves feasibility while enabling weight-analytic inequalities. The paper also highlights a link between the BE&S problem and generalized Ramsey numbers $\mathrm{GR}(n,p,q_{quad})$, providing asymptotic consequences such as $\lim_{n\to\infty}\mathrm{GR}(n,18,146)/n^2=5/12$. Overall, the results complete the picture for $k=8$ with tight upper and lower bounds and bolster the merge-weight methodology for extremal hypergraph problems.
Abstract
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.