  For fixed integers $r>k\ge 2,e\ge 3$, let $f_r(n,er-(e-1)k,e)$ be the maximum
number of edges in an $r$-uniform hypergraph in which the union of any $e$
distinct edges contains at least $er-(e-1)k+1$ vertices. A classical result of
Brown, Erd\H{o}s and S\'os in 1973 showed that
$f_r(n,er-(e-1)k,e)=\Theta(n^k).$ The degenerate Tur\'an density is defined to
be the limit (if it exists)
$$\pi(r,k,e):=\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}.$$
Extending a recent result of Glock for the special case of $r=3,k=2,e=3$, we
show that
$$\pi(r,2,3):=\lim_{n\rightarrow\infty}\frac{f_r(n,3r-4,3)}{n^2}=\frac{1}{r^2-r-1}$$
for arbitrary fixed $r\ge 4$. For the more general cases $r>k\ge 3$, we show
that
$$\frac{1}{r^k-r}\le\liminf_{n\rightarrow\infty}\frac{f_r(n,3r-2k,3)}{n^k}\le\limsup_{n\rightarrow\infty}\frac{f_r(n,3r-2k,3)}{n^k}\le
\frac{1}{k!\binom{r}{k}-\frac{k!}{2}}.$$
  The main difficulties in proving these results are the constructions
establishing the lower bounds. The first construction is recursive and purely
combinatorial, and is based on a (carefully designed) approximate induced
decomposition of the complete graph, whereas the second construction is
algebraic, and is proved by a newly defined matrix property which we call {\it
strongly 3-perfect hashing}.
