Sieving very thin sets of primes

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$
contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.