Sparse hypergraphs: new bounds and constructions

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.