  Let $f_r(n,v,e)$ denote the maximum number of edges in an $r$-uniform
hypergraph on $n$ vertices, in which the union of any $e$ distinct edges
contains at least $v+1$ vertices. The study of $f_r(n,v,e)$ was initiated by
Brown, Erd{\H{o}}s and S{\'o}s more than forty years ago. In the literature,
the following conjecture is well known.
  Conjecture: $n^{k-o(1)}<f_r(n,er-(e-1)k+1,e)=o(n^k)$ holds for all fixed
integers $r>k\ge 2$ and $e\ge 3$ as $n\rightarrow\infty$.
  For $r=3, e=3, k=2$, the bound $n^{2-o(1)}<f_3(n,6,3)=o(n^2)$ was proved by
the celebrated (6,3)-theorem of Ruzsa and Szemer{\'e}di. In this paper, we add
more evidence for the validity of the conjecture. On one hand, using the
hypergraph removal lemma we show that the upper bound part of the conjecture is
true for all fixed integers $r\ge k+1\ge e\ge3$.
  On the other hand, using tools from additive number theory we present several
constructions showing that the lower bound part of the conjecture is true for
$r\ge3$, $k=2$ and $e=4,5,7,8$.
  Prior to our results, all known constructions that match the conjectured
lower bound satisfy either $r=3$ or $e=3$. Our constructions are the first ones
in the literature that break this barrier.
