An $r$-graph is called $(v,e)$-free if the union of any $e$ distinct edges of it contains at least $v+1$ vertices. Let $f_r(n,v,e)$ denote the maximum number of edges of a $(v,e)$-free $r$-graph on $n$ vertices. The study of the function $f_r(n,v,e)$ was initiated by Brown, Erdös and Sós in 1973. In the literature, the following two conjectures are well-known.

Conjecture 1. For all fixed $e\ge 3$ and $r\ge k+1\ge3$, $n^{k-o(1)}<f_r(n,er-(e-1)k+1,e)=o(n^k)$.

Conjecture 2. For all fixed $e\ge 3$ and $r\ge k+1\ge3$, the limit $\lim_{n\rightarrow\infty} f_r(n,er-(e-1)k,e)/n^{k}$ always exist.

In this talk, we will discuss some recent progress on these two conjectures.