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Suppose P is a set of primes, such that for every p in P, every prime factorof p-1 is also in P. If P does not contain all primes, we apply a new sievemethod to show that the counting function of P is O(x^{1-c}) for some c>0,where c depends only on the smallest prime not in P. Our proof makes use ofresults connected with Artin's primitive root conjecture.

Sieving by very thin sets of primes, and Pratt trees with missing primes