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 eβ₯3, what is the smallest integer d=d(e) so that
f(n,e+d,e)=o(n2)? 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 eβ₯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+βlog2βeβ,e)=o(n2). 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). $