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.

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appeared in J. London. Math. Soc. 48 (1993), no 2 193--203.