We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP₂.

In this paper we prove that a degree a is array nonrecursive (ANR) if and only if every degree b≥a is r.e. in and strictly above another degree (RRE). This result will answer some questions in [ASDWY]. We also deduce an interesting corollary that every n-REA degree has a strong minimal cover if and only if it is array recursive.

We present some abstract theorems showing how domination properties equivalent to being $\overline{GL}_2$ or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang and Yu [2009] that every degree above any in $\overline{GL}_2$ is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below $\mathbf{ANR}$ degrees also reveal a new level of uniformity in these types of results.

Let $A$ be a non-empty set. $A$ set $S \subseteq \mathscr{P}(A)$ is said to be stationary in $\mathscr{P}(A)$ if for every $f: [A]^{< \omega} \to A$ there exists $ x \in S$ such that $x \neq A$ and $f"[x]^{<\omega}$. In this paper we prove the following: For an uncountable cardinal ? and a stationary set S in $\mathscr{P}(\lambda)$, if there is a regular uncountable cardinal $k \leq \lambda$ such that $\{x \in S: x \cap k \in k\}$ is stationary, then S can be split into $k$ disjoint stationary subsets.

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n-1.

We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ¹₁ equivalence relations on hyperarithmetical subsets of ω.

We show that every splitting of 0_e' in the local structure of the enumeration degrees, 𝔊_e, contains at least one low-cuppable member. We apply this new structural property to show that the classes of all 𝓚-pairs in 𝔊_e, all downwards properly Σ⁰₂ enumeration degrees and all upwards properly Σ⁰₂ enumeration degrees are first order definable in 𝔊_e.

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y)≤ c d(x,z) for all x<y<z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon)≥𝔪_σ-linked, but non(Mon)<𝔪_σ-centered is consistent. Also cov(Mon)<𝔠 and cof(𝒩)<cov(Mon) are consistent.

A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5].

Let T₁, T₂ be countable first-order theories, M_i ⊨ T_i, and 𝒟 any regular ultrafilter on λ ≥ ℵ₀. A longstanding open problem of Keisler asks when T₂ is more complex than T₁, as measured by the fact that for any such λ, 𝒟, if the ultrapower (M₂)^λ/𝒟 realizes all types over sets of size ≤ λ, then so must the ultrapower (M₁)^λ/𝒟. In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP₂ theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular decompositions) remaining bounded away from 0,1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP₂ theory is flexible.

We investigate Borel reducibility between equivalence relations E(X;p)=X^ℕ/ℓ_p(X)'s where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L_r;p)'s and E(c₀;p)'s for r,p∈[1,+∞).

We also answer a problem presented by Kanovei in the affirmative by showing that C(ℝ⁺)/C₀(ℝ⁺) is Borel bireducible to ℝ^ℕ/c₀.

We show that the variety of weakly representable relation algebras is neither canonical nor closed under Monk completions.

This work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.

We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵ_ω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

Dynamic Topological Logic (𝒟𝒯ℒ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X,f〉 where X is a topological space and f: X→ X a continuous function.

In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.

We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.

We prove that in the Polish group of isometries of the Baire space the collection of n-th powers is non-Borel. We also prove that in the Polish space of trees on ℕ the collection of trees that have an automorphism under which every node has order exactly n is non-Borel.

We investigate the isomorphism types of combinatorial geometries arising from Hrushovski's flat strongly minimal structures and answer some questions from Hrushovski's original paper.

We consider ωⁿ-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωⁿ for some integer n≥1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω²-automatic (resp. ωⁿ-automatic for n>2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ωⁿ-automatic boolean algebras, n≥2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a Σ₂¹-set nor a Π₂¹-set. We obtain that there exist infinitely many ωⁿ-automatic, hence also ω-tree-automatic, atomless boolean algebras ℬ_n, n≥1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].