Let
f
(
r
)
(
n
;
s
,
k
)
f^{(r)}(n;s,k)
be the maximum number of edges of an
r
r
-uniform hypergraph on
n
n
vertices not containing a subgraph with
k
k
edges and at most
s
s
vertices. In 1973, Brown, Erdős, and Sós conjectured that the limit
lim
n
→
∞
n
−
2
f
(
3
)
(
n
;
k
+
2
,
k
)
\begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*}
exists for all
k
k
and confirmed it for
k
=
2
k=2
. Recently, Glock showed this for
k
=
3
k=3
. We settle the next open case,
k
=
4
k=4
, by showing that
f
(
3
)
(
n
;
6
,
4
)
=
(
7
36
+
o
(
1
)
)
n
2
f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right )n^2
as
n
→
∞
n\to \infty
. More generally, for all
k
∈
{
3
,
4
}
k\in \{3,4\}
,
r
≥
3
r\ge 3
and
t
∈
[
2
,
r
−
1
]
t\in [2,r-1]
, we compute the value of the limit
lim
n
→
∞
n
−
t
f
(
r
)
(
n
;
k
(
r
−
t
)
+
t
,
k
)
\lim _{n\to \infty } n^{-t}f^{(r)}(n;k(r-t)+t,k)
, which settles a problem of Shangguan and Tamo.