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, Erd\H{ o}s and S\'os 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, K\"{ u}hn, Lichev, Pikhurko, R\"odl 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.
Categories: math.CO