We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the ℵ0-categorical case we show that this closure is part of the algebraic closure.

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.

The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.

The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.

It is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .

We consider the monadic second-order theory of linear order. For the sake of brevity, linearly ordered sets will be called chains.

Let = ⟨A <⟩ be a chain. A formula ø(t) with one free individual variable t defines a point-set on A which contains the points of A that satisfy ø(t). As usually we identify a subset of A with its characteristic predicate and we will say that such a formula defines a predicate on A.

A formula (X) one free monadic predicate variable defines the set of predicates (or family of point-sets) on A that satisfy (X). This family is said to be definable by (X) in A. Suppose that is a subchain of = ⟨B, <⟩. With a formula (X, A) we associate the following family of point-sets (or set of predicates) {P : P ⊆ A and (P, A) holds in } on A. This family is said to be definable by in with at the background.

Note that in such a definition bound individual (respectively predicate) variables of range over B (respectively over subsets of B). Hence, it is reasonable to expect that the presence of a background chain allows one to define point sets (or families of point-sets) on A which are not definable inside .

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.