Mathematics

We study forcing notions similar to the Cohen forcing, whichadd some struc-tures in given first-order language. These structures can beseen as versions of uncountable Fraïssé limits with finite conditions. Among them, we are primarily interested in linear orders.

Let K=⟨R,δ⟩ be a closed ordered differential field, in the sense of M. Singer, and C its field of constants. In this note, we prove that, for sets definable in the pair M=⟨R,C⟩ , the δ -dimension and the large dimension coincide. As an application, we characterize the definable sets in K that are internal to C as those sets that are definable in M and have δ -dimension 0 . We further show that, for sets definable in K , having δ -dimension 0 does not generally imply co-analyzability in C (in contrast to the case of transseries). We also point out that the coincidence of dimensions also holds in the context of differentially closed fields and in the context of transseries.

We prove several consistency results in choiceless set theory ZF+DC regarding countable chromatic numbers of various algebraic hypergraphs on Euclidean spaces.

We discuss some properties of Cohen and random reals. We show that they belong to any definable partition regular family, and hence they satisfy most "largeness" properties studied in Ramsey theory. We determine their position in the Mahler's classification of the reals and using it, we get some information about Liouville numbers. We also show that they are wild in the sense of o-minimality, i.e., they define the set of integers.

We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.

It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes. The extent to which resurrection and maximality overlap is explored via the local maximality principle.

We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, that is, the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic, the equational theory of *-continuous action lattices. Namely, we prove that the former is Σ 0 1 -complete and the latter is ? 0 1 -complete. Thus, the situation is the same as in the more well-studied non-commutative case. The methods used, however, are different: we encode infinite and circular computations of counter (Minsky) machines.

If κ is regular and 2 <κ ≤ κ + , then the existence of a weakly presaturated ideal on κ + implies □ ∗ κ . This partially answers a question of Foreman and Magidor about the approachability ideal on ω 2 . As a corollary, we show that if there is a presaturated ideal I on ω 2 such that P( ω 2 )/I is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

This paper provides a full characterization for when the expansion of a complete o-minimal theory by a unary predicate that picks out a divisible dense and codense subgroup has a model companion. This result is motivated by criteria and questions introduced in the recent works concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. The focus of this paper is establishing the companionability dividing line in the o-minimal setting because this allows us to provide a full and geometric characterization. Examples are included both in which the predicate is an additive subgroup, and where it is a multiplicative subgroup. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the "base" o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP 2 , though there are also examples for which some or all three of those properties hold.

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. In particular, we show the existence of non-quantitative lower bounds on the number of arithmetic progressions of length 3 for subsets of small doubling without involutions in arbitrary abelian groups.