We let
G
(r)(
n,m) denote the set of
r-uniform hypergraphs with
n vertices and
m edges, and
f
(r)(
n,p,s) is the smallest m such that every member of
G
(r)(
n,m) contains amember of
G
(r)(
p,s). In this paper we are interested in fixed values
r,p and
s for which
f
(r)(
n,p,s) grows quadratically with
n. A probabilistic construction of Brown, Erd
$
f^{{{\left( r \right)}}} {\left( {n,s{\left( {r - 2} \right)} + 2 + {\left\lfloor {\log _{2} s} \right\rfloor },s} \right)} = o{\left( {n^{2} } \right)}.
$
f^{{{\left( r \right)}}} {\left( {n,s{\left( {r - 2} \right)} + 2 + {\left\lfloor {\log _{2} s} \right\rfloor },s} \right)} = o{\left( {n^{2} } \right)}.