  A $\left(n,\ell,\gamma\right)$-sharing set family of size $m$ is a family of
sets $S_1,\ldots,S_m\subseteq [n]$ s.t. each set has size $\ell$ and each pair
of sets shares at most $\gamma$ elements. We let $m\left(n,\ell,\gamma\right)$
denote the maximum size of any such set family and we consider the following
question: How large can $m\left(n,\ell,\gamma\right)$ be?
$\left(n,\ell,\gamma\right)$-sharing set families have a rich set of
applications including the construction of pseudorandom number generators and
usable and secure password management schemes. We analyze the explicit
construction of Blocki et al using recent bounds on the value of the $t$'th
Ramanujan prime. We show that this explicit construction produces a
$\left(4\ell^2\ln 4\ell,\ell,\gamma\right)$-sharing set family of size $\left(2
\ell \ln 2\ell\right)^{\gamma+1}$ for any $\ell\geq \gamma$. We also show that
the construction of Blocki et al can be used to obtain a weak
$\left(n,\ell,\gamma\right)$-sharing set family of size $m$ for any $m >0$.
These results are competitive with the inexplicit construction of Raz et al for
weak $\left(n,\ell,\gamma\right)$-sharing families. We show that our explicit
construction of weak $\left(n,\ell,\gamma\right)$-sharing set families can be
used to obtain a parallelizable pseudorandom number generator with a low memory
footprint by using the pseudorandom number generator of Nisan and Wigderson. We
also prove that $m\left(n,n/c_1,c_2n\right)$ must be a constant whenever $c_2
\leq \frac{2}{c_1^3+c_1^2}$. We show that this bound is nearly tight as
$m\left(n,n/c_1,c_2n\right)$ grows exponentially fast whenever $c_2 >
c_1^{-2}$.
