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A (n, ℓ, γ)-sharing set family of size m is a family of sets S 1 , . . . , S m ⊆ [n] s.t. each set has size ℓ and each pair of sets shares at most γ elements. We let m (n, ℓ, γ) denote the maximum size of any such set family and we consider the following question: How large can m (n, ℓ, γ) be? (n, ℓ, γ)-sharing set families have a rich set of applications including the construction of pseudorandom number generators[NW94] and usable and secure password management schemes [BBD13]. We analyze the explicit construction of Blocki et al [BBD13] using recent bounds [Son09] on the value of the t'th Ramanujan prime [Ram19]. We show that this explicit construction produces a 4ℓ 2 ln 4ℓ, ℓ, γ -sharing set family of size (2ℓ ln 2ℓ)γ+1 for any ℓ ≥ γ. We also show that the construction of Blocki et al [BBD13] can be used to obtain a weak (n, ℓ, γ)-sharing set family of size m for any m > 0. These results are competitive with the inexplicit construction of Raz et al [RRV99] for weak (n, ℓ, γ)-sharing families. We show that our explicit construction of weak (n, ℓ, γ)-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[NW94]. We also prove that m (n, n/c 1 , c 2 n) must be a constant whenever c 2 ≤ 2 c 3 1 +c 2 1 . We show that this bound is nearly tight as m (n, n/c 1 , c 2 n) grows exponentially fast whenever c 2 > c −2 1 .

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