We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.

We study the definability of ultrafilter bases on \ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct \ P-point and Q-point bases. We also show that the existence of a \ ultrafilter is equivalent to that of a \ ultrafilter base, for \. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.

Todorčević (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses (...) Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$. (shrink)

A cofinitary group is a subgroup of Sym(ℕ) where all nonidentity elements have finitely many fixed points. A maximal cofinitary group is a cofinitary group, maximal with respect to inclusion. We show that a maximal cofinitary group cannot have infinitely many orbits. We also show, using Martin's Axiom, that no further restrictions on the number of orbits can be obtained. We show that Martin's Axiom implies there exist locally finite maximal cofinitary groups. Finally we show that there exists a uniformly (...) computable sequence of permutations generating a cofinitary group whose isomorphism type is not computable. (shrink)

If F ⊆ NN is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ NN. no member of which is covered by finitely many functions from F, there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality (...) condition. (shrink)

Descriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. (...) In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for $\boldsymbol {\Sigma ^{1}_{2}}$ sets of reals. (shrink)

We show that the non-existence of mad families is equiconsistent with \, answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that \ “there is no maximal independent set in G” is equiconsistent with \ “there exists an inaccessible cardinal”.

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with.

We show that the existence of a ${\Pi^1_1}$ -definable mad family is consistent with the existence of a ${\Delta^{1}_{3}}$ -definable well-order of the reals and ${\mathfrak{b}=\mathfrak{c}=\aleph_3}$.

We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

We study the problem of existence of maximal chains in the Turing degrees. We show that: 1. ZF+DC+"There exists no maximal chain in the Turing degrees" is equiconsistent with ZFC+"There exists an inaccessible cardinal"; 2. For all a ∈ 2ω.(ω₁)L[a] = ω₁ if and only if there exists a $\Pi _{1}^{1}[a]$ maximal chain in the Turing degrees. As a corollary, ZFC + "There exists an inaccessible cardinal" is equiconsistent with ZFC + "There is no (bold face) $\utilde{\Pi}{}_{1}^{1}$ maximal chain of (...) Turing degrees". (shrink)

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, (...) and gives partial answers to questions of Terada and Medvedev. (shrink)

We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, (...) and it was proved that Σ21-Kσ-regularity implies Σ21-u-regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u-regularity uses a game argument it was clear that determinacy implies u-regularity of all sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u-regularity, namely it holds in Solovay's model. Finally we show that forcing with uniform trees is equivalent to Laver forcing. (shrink)

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...) of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)