We show how to build various models of first-order theories, which also have properties like: tree with only definable branches, atomic Boolean algebras or ordered fields with only definable automorphisms. For this we use a set-theoretic assertion, which may be interesting by itself on the existence of quite generic subsets of suitable partial orders of power λ + , which follows from ♦ λ and even weaker hypotheses . For a related assertion, which is equivalent to the morass see Shelah (...) and Stanley [16]. The various specific constructions serve also as examples of how to use this set-theoretic lemma. We apply the method to construct rigid ordered fields, rigid atomic Boolean algebras, trees with only definable branches; all in successors of regular cardinals under appropriate set- theoretic assumptions. So we are able to answer the following algebraic question. Saltzman's Question . Is there a rigid real closed field, which is not a subfield of the reals? (shrink)

THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...) B is an atomic ℵ 1 -saturated infinite BA, ψ ε L ω 1ω and $\langle B, \operatorname{Aut} (B)\rangle \models\psi$ then there is B 1 such that $|\operatorname{Aut}(B_1)| \leq |B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut}(B_1)\rangle\models\psi$ . In particular if B is 1-homogeneous so is B 1 . (b) (a) holds for B = P(ω) even if we assume only CH. (shrink)