Sets of prime numbers satisfying a divisibility condition

We study sets P of k primes that satisfy the condition gcd(Pi A - Pi B, Pi P) = 1 whenever A and B are disjoint non-empty subsets of P. It is known that such sets of primes exist for all positive integers k. It is of interest to know the asymptotic behavior of n(k), the smallest natural number that is the product of k such primes. In this paper we derive asymptotic bounds for n(k). (C) 1996 Academic Press, Inc.