S. Shelah, L. Soukup:
The existence of large omega_1-homogeneous
but not omega-homogeneous
permutation groups is consistent with ZFC + GCH
Denote by Perm(lambda) the group
of all permutations of a cardinal lambda.
A subgroup G of Perm(lambda)
is called kappa-homogeneous iff
for all X,Y\in [lambda]^kappa
there is a g\in G with g''X=Y. We show that if either
(i) diamond^+ holds
and we add omega_1 Cohen reals to the ground model,
or (ii) we add 2^{omega_1} Cohen reals to the ground model,
then in the generic extension
for each lambda>=omega_2 there is an
omega_1-homogeneous subgroup
of Perm(lambda) which is
not omega-homogeneous.
Downloading the paper
appeared in J. London. Math. Soc. 48 (1993), no 2 193--203.