The Existence of Large ω₁-Homogeneous But Not ω-Homogeneous Permutation Groups is Consistent with Zfc+Gch

Denote by Perm (λ) the group of all permutations of a cardinal λ. A subgroup G of Perm (λ) is called K-homogeneous if and only if for all X, Y € [λ]^K there is a g € G with g″X = Y. We show that if either (i) ⋄⁺ holds and we add ω₁ Cohen reals to the ground model, or (ii) we add 2^ω₁ Cohen reals to the ground model, then in the generic extension for each λ ≥ ω₂ there is an ω₁-homogeneous subgroup of Perm (λ) which is not ω-homogeneous.