A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness

The only real relational structures of scale type (m, n) with 1 ≤ m ≤ n < ∞ are of scale type (1, 1), (1, 2), and (2, 2), and so are conjugate to structures whose automorphism groups are subgroups of the affines containing the group of translations. All real relational structures of scale type (0, n) with n < ∞ have automorphism groups abstractly isomorphic to; moreover, each contains subchain of order type theta, invariant under the action of the automorphism group, s.t. the action of the automorphisms on this subchain is conjugate to that of a sub-group of the affines.