Let K be a number field with ring of integers OK, and let
{fk}k∈N⊆OK[x] be a sequence of monic
polynomials such that for every n∈N, the composition
f(n)=f1∘f2∘…∘fn is irreducible. In this paper we
show that if the size of the Galois group of f(n) is large enough (in a
precise sense) as a function of n, then the set of primes p⊆OK such that every f(n) is irreducible modulo
p has density zero. Moreover, we prove that the subset of
polynomial sequences such that the Galois group of f(n) is large enough
has density 1, in an appropriate sense, within the set of all polynomial
sequences.Comment: Comments are welcome