Abstract Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$ . We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$ . Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$ . Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$ .

Abstract AssumeGis a group definable in a modelMof a stable theoryT. We prove that the semigroupSG(M) of completeG-types overMis an inverse limit of some semigroups type-definable inMeq. We prove that the maximal subgroups ofSG(M) are inverse limits of some definable quotients of subgroups ofG. We consider the powers of types in the semigroupSG(M) and prove that in a way every type inSG(M) is profinitely many steps away from a type in a subgroup ofSG(M).

Abstract We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

Abstract We consider a small supersimple theory with a property (CS) (close to stability). We prove that if in such a theoryTthere is a typep∈S(A) (whereAis finite) withSU(p) = 1 and infinitely many extensions overacleq(A), then inTthere is a modular such type. Also, ifTis supersimple with (CS) andp∈S(∅) is isolated,SU(p) = 1 andphas infinitely many extensions overacleq(∅), thenpis modular.

Abstract We propose a model-theoretic framework for investigating profinite groups. Within this framework we define and investigate small profinite groups. We consider the question if any small profinite group has an open abelian subgroup.

Abstract AssumeTis a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a typep, in case whenpis a complete type over a finite set of parameters. We show that for any flat Morley sequenceQthere is a modelMofTwhich isτ-atomic over {Q}. When additionallyThas few countable models and is 1-based, we prove that withinMthere is an infinite Morley sequenceI, withI⊂ dcl(Q), such thatMis prime overI.

Abstract AssumeTis a superstable theory with < 2ℵ0countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic typep*of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated withp*is a meager typep. We prove thatpis determined byp*up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization ofp. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.

Abstract Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on T⌈Φ of naming finitely many elements of a model of T. We consider the cases of T⌈Φ which is ω-stable or superstable of finite rank. In these cases we prove that if T has countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If T⌈Φ is ω-stable and (bounded, 1-based or of finite rank) with , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.

Abstract This is a continuation of [N2]. We find a Borel definition of Q-isolation. We pursue a topological and Scott analysis of pseudotypes on S(Q).

Abstract We try to count the number of countable models M of T with a fixed set Q = Φ(M) of realizations of a type Φ. Also, for stable T, we define an ordinal rank measuring multiplicity of types, with additivity properties similar to those of U-rank.

In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1]. Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular T is always a countable complete theory in the language L. We consider all models of T and all sets of parameters subsets of the monster model ℭ, which is very saturated. Ln(A) denotes the Lindenbaum-Tarski algebra of formulas with parameters from A and n free variables. We omit n in Ln(A) when n = 1 or when it is clear from the context what n is. If φ, ψ ∈ L(A) are consistent then we say that φ is below ψ if ψ⊢ψ. For a type p and a set A ⊆ ℭ, p(A) is the set of tuples of elements of A which satisfy p. Formulas are special cases of types. We say that a type p is isolated over A if, for some φ( ) ∈ L(A), φ( ) ⊢ p(x), i.e. φ isolates p. For a formula φ, [φ] denotes the class of types which contain φ. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M]. Throughout, we work in ZFC. and denote (countable) transitive models of ZFC. cov K is the minimal number of meager sets covering the real line R. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].

The problem mentioned in the title has already been investigated by J. Baumgartner, J. Stern, A. Miller and many others (see [2] and [5]). We prove here some generalizations of theorems of Miller and Stern from [2] and [5]. We use standard set-theoretical notation. Let One can check that in the above definition we can replace “compact subset of ωω” by “closed nowhere dense subset of ω2” or “Fσ and meager subset of ω2” (as any Fσ subset of ω2 can be presented as a disjoint countable union of compact sets). For functions f, g ϵ ωω we define f ≼ g if for all but finitely many n ϵ ω we have f(n) ≤ g(n). Let denote the least cardinality of a family A ⊆ ωω such that for any f ϵ ωω there is g ϵ A for which f ≼ g. It is easy to see that ≤ κω ≤ κ1. If f ϵ ωω then let ≼(f) = {h ϵωω: h ≼ f}. We find an axiom which implies = ω1 → κ1 = ω1, and which can be preserved by any ccc notion of forcing of “small cardinality”. We construct also in a generic model many partitions of ωω into compact sets preserved not only by any random real extension, but also by Sacks' notion of forcing. This shows that from some point of view Miller's modification of Sacks' forcing (from [2]) is the “minimal” one able to destroy a partition of ωω into compact sets.

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets ofR. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types. It is well known that assuming MA we can omit < nonisolated types. But MA is rather a strong axiom. We prove that in order to be able to omit < nonisolated types it is sufficient to assume that the real line cannot be covered by less than meager sets; and this is in fact the weakest possible condition. It is worth pointing out that by means of forcing we can easily obtain the model of ZFC in whichRcannot be covered by < meager sets. It suffices to add to the ground model Cohen generic reals. We also formulate similar results for omitting pairwise contradictory types. It turns out that from some point of view it is much more difficult to find the family of pairwise contradictory types which cannot be omitted by a model ofT, than to find such a family of possibly noncontradictory types. Moreover, for any two countable theoriesT1,T2without prime models, the existence of a family ofκtypes which cannot be omitted by a model ofT1is equivalent to the existence of such a family forT2. This means that from the point of view of omitting types all theories without prime models are identical. Similar results hold for omitting pairwise contradictory types.