  Let $f(n, v, e)$ denote the maximum number of edges in a 3-uniform hypergraph
on $n$ vertices which does not contain $v$ vertices spanning at least $e$
edges. A central problem in extremal combinatorics, famously posed by Brown,
Erd\H{o}s and S\'os in 1973, asks whether $f(n, e+3, e)=o(n^2)$ for every $e
\ge 3$. A classical result of S\'ark\"ozy and Selkow states that $f(n,
e+\lfloor \log_2 e\rfloor+2, e)=o(n^{2})$ for every $e \ge 3$. This bound was
recently improved by Conlon, Gishboliner, Levanzov and Shapira.
  Motivated by applications to other problems, Gowers and Long made the
striking conjecture that $f(n, e+4, e)=O(n^{2-\varepsilon})$ for some
$\varepsilon=\varepsilon(e)>0$. Conlon, Gishboliner, Levanzov and Shapira, and
later, Shapira and Tyomkyn reiterated the following approximate version of this
problem. What is the smallest $d(e)$ for which $f(n, e+d(e),
e)=O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$? In this
paper, we prove that for each $e\geq 3$ we have $f(n, e+\lfloor \log_2 e\rfloor
+38, e)=O(n^{2-\varepsilon})$ for some $\varepsilon>0$. This shows that one can
already obtain power saving near the S\'ark\"ozy-Selkow bound at the cost of a
small additive constant.
