  Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq
k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform
hypergraph on $n$ vertices can have under the condition that any collection of
$i$ edges, span at least $i$ vertices for all $1 \leq i \leq k$. We are
interested in the asymptotic nature of $m(n, r, k)$ for fixed $r$ and $k$ as $n
\rightarrow \infty$. This problem is related to the forbidden hypergraph
problem introduced by Brown, Erd\H{o}s, and S\'os and very recently discussed
in the context of combinatorial batch codes. In this short paper we obtain the
following results. {enumerate}[(i)] Using a result due to Erd\H{o}s we are able
to show $m(n, k, r) = o(n^r)$ for $7\leq k$, and $3 \leq r \leq k-1-\lceil\log
k \rceil$. This result is best possible with respect to the upper bound on $r$
as we subsequently show through explicit construction that for $6 \leq k$, and
$k-\lceil \log k \rceil \leq r \leq k-1, m(n, r, k) = \Theta(n^r)$.
  This explicit construction improves on the non-constructive general lower
bound obtained by Brown, Erd\H{o}s, and S\'os for the considered parameter
values. For 2-uniform CBCs we obtain the following results. {enumerate} We
provide exact value of $m(n, 2, 5)$ for $n \geq 5$. Using a result of
Lazebnik,et al. regarding maximum size of graphs with large girth, we improve
the existing lower bound on $m(n, 2, k)$ ($\Omega(n^{\frac{k+1}{k-1}})$) for
all $k \geq 8$ and infinitely many values of $n$. We show $m(n, 2, k) =
O(n^{1+\frac{1}{\lfloor\frac{k}{4}\rfloor}})$ by using a result due to Bondy
and Simonovits, and also show $m(n, 2, k) = \Theta(n^{3/2})$ for $k = 6, 7, 8$
by using a result of K\"{o}vari, S\'os, and Tur\'{a}n.
