  The current paper answers an open question of abs/1007.2426
  We say that a countable model M characterizes an infinite cardinal kappa, if
the Scott sentence of M has a model in cardinality kappa, but no models in
cardinality kappa plus. If M is linearly ordered by <, we will say that the
linear ordering (M,<) characterizes kappa. It is known that if kappa is
characterizable, then kappa plus is characterizable by a linear ordering. Also,
if kappa is characterizable by a dense linear ordering with an increasing
sequence of size kappa, then 2^kappa is characterizable. We show that if kappa
is homogeneously characterizable, then kappa is characterizable by a dense
linear ordering, while the converse fails.
  The main theorems are: 1) If kappa>2^lambda is a characterizable cardinal,
lambda is characterizable by a dense linear ordering and lambda is the least
cardinal such that kappa^lambda>kappa, then kappa^lambda is also
characterizable, 2) if aleph_alpha and kappa^(aleph_alpha) are characterizable
cardinals, then the same is true for kappa^(aleph_(alpha+beta)), for all
countable beta. Combining these two theorems we get that if
kappa>2^(aleph_alpha) is a characterizable cardinal, aleph_alpha is
characterizable by a dense linear ordering and aleph_alpha is the least
cardinal such that kappa^(aleph_alpha)>kappa, then for all beta<alpha+omega_1,
kappa^(aleph_beta) is characterizable. Also if kappa is a characterizable
cardinal, then kappa^(aleph_alpha) is characterizable, for all countable alpha.
