  We prove that the irreducible affine Coxeter groups are first-order rigid and
deduce from this that they are profinitely rigid in the absolute sense. We then
show that the first-order theory of any irreducible affine Coxeter group does
not have a prime model. Finally, we prove that universal Coxeter groups of
finite rank are homogeneous, and that the same applies to every hyperbolic (in
the sense of Gromov) one-ended right-angled Coxeter group.
