  Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph
not containing $e$ edges spanned by at most $v$ vertices. One of the most
influential open problems in extremal combinatorics then asks, for a given
number of edges $e \geq 3$, what is the smallest integer $d=d(e)$ so that
$f(n,e+d,e) = o(n^2)$? This question has its origins in work of Brown,
Erd\H{o}s and S\'os from the early 70's and the standard conjecture is that
$d(e)=3$ for every $e \geq 3$. The state of the art result regarding this
problem was obtained in 2004 by S\'{a}rk\"{o}zy and Selkow, who showed that
$f(n,e + 2 + \lfloor \log_2 e \rfloor,e) = o(n^2)$. The only improvement over
this result was a recent breakthrough of Solymosi and Solymosi, who improved
the bound for $d(10)$ from 5 to 4. We obtain the first asymptotic improvement
over the S\'{a}rk\"{o}zy--Selkow bound, showing that $$ f(n, e + O(\log e/
\log\log e), e) = o(n^2). $$
