  We define a new class of sets -- stable sets -- of primes in number fields.
For example, Chebotarev sets $P_{M/K}(\sigma)$, with $M/K$ Galois and $\sigma
\in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary
small) Dirichlet density and generalize sets with density 1 in the sense that
arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem,
the Riemann's existence theorem, etc. hold for them. Geometrically this allows
to give examples of infinite sets $S$ with arbitrary small positive density
such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(\pi,1)$ (for all $p$
simultaneous).
