Sparse hypergraphs: New bounds and constructions

More than forty years ago, Brown, Erdős and Sos introduced the function $f_r(n,v,e)$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which does not contain $e$ edges spanned by $v$ vertices. In other words, in such a hypergraph the union of arbitrary $e$ edges contains at least $v+1$ vertices. In the literature, the following conjecture is well-known. Conjecture: $n^{k-o(1)} k\ge 2$, $e\ge 3$. Note that for $r=3, e=3, k=2$, this conjecture was solved by the famous Ruzsa-Szemer{e}di's (6,3)-theorem. In this paper, we add more evidence for the validity of this conjecture. On one hand, we use the hypergraph removal lemma to prove that the right hand side of the conjecture is true for all fixed integers $r\ge k+1\ge e\ge3$. Our result implies all known upper bounds which match the conjectured magnitude. On the other hand, we present several constructive results showing that the left hand side of the conjecture is true for $r\ge3$, $k=2$ and $e=4,5,7,8$. All previous constructive results meeting the conjectured lower bound satisfy either $r=3$ or $e=3$. Our constructions are the first ones which break this barrier.