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For
$G$
a Polish group, we consider
$G$
-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.
The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action
$\unicode[STIX]{x1D6FC}$
is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by
$\unicode[STIX]{x1D6FC}$
is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When
$\unicode[STIX]{x1D6FC}$
is the shift action, we characterize the class of problems for which
$\unicode[STIX]{x1D6FC}$
has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.
An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and
$n$
-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the
$2$
-sphere. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of
$\mathsf{PSL}_{2}(\mathbb{Z})$
on
$\mathsf{P}^{1}(\mathbb{R})$
. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.
Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let
$f:{2^\omega } \to {X_L}$
be a hyperarithmetic function such that for all
$x,y \in {2^\omega }$
, if
$x{ \equiv _h}y$
then
$f\left( x \right) \cong f\left( y \right)$
. One of the following two properties must hold. (1) The Scott rank of f (0) is
$\omega _1^{CK} + 1$
. (2) For all
$x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$
.
We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.
We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that
$\Pi _{n + 1}^1 $
-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies
$$A = R$$
and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.
We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in
$^R R\Pi _1^1 $
or with σ-projective payoff.
We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.
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 prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets
$\mathbb{P}_{emb} $
equipped with the order induced by homomorphisms is embedded into the Wadge order on the
$\Delta _2^0 $
-degrees of the Scott domain. We then show that
$\mathbb{P}_{emb} $
admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the
$\Delta _2^0 $
-degrees of the Scott domain.
We study the behavior of the game operator
$$
on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes
${\bf{\Gamma }}$
for which the class has the substitution property. An effective variation of these results shows that for all
$1 \le \eta < \omega _1^{{\rm{CK}}}$
and
$2 \le \xi < \omega _1^{{\rm{CK}}}$
, is a Spector class while is not.
Duparc introduced a two-player game for a function f between zero-dimensional Polish spaces in which Player II has a winning strategy iff f is of Baire class 1. We generalize this result by defining a game for an arbitrary function f : X → Y between arbitrary Polish spaces such that Player II has a winning strategy in this game iff f is of Baire class 1. Using the strategy of Player II, we reprove a result concerning first return recoverable functions.
We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.
Given a compact Polish space E and the hyperspace of its compact subsets
${\cal K}\left( E \right)$
, we consider Gδσ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of
${\cal K}\left( E \right)$
; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set that recognises a compact subset of E as being “small” precisely when it is in I, and recognises a Gδ subset of E as being “small” precisely when it is covered by countably many compact sets from I.
Using a nonLaver modification of Uri Abraham’s minimal
$\Delta _3^1$
collapse function, we define a generic extension
$L[a]$
by a real a, in which, for a given
$n \ge 3$
,
$\left\{ a \right\}$
is a lightface
$\Pi _n^1 $
singleton, a effectively codes a cofinal map
$\omega \to \omega _1^L $
minimal over L, while every
$\Sigma _n^1 $
set
$X \subseteq \omega $
is still constructible.
We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of
${\bf{\Sigma }}_1^0 \times {\bf{\Sigma }}_\xi ^0$
sets, or by a
${\bf{\Pi }}_1^0 \times {\bf{\Pi }}_\xi ^0$
set.
We prove the following two basis theorems for
${\rm{\Sigma }}_2^1$
-sets of reals:
(1)Every nonthin
${\rm{\Sigma }}_2^1$
-set has a perfect
${\rm{\Delta }}_2^1$
-subset if and only if it has a nonthin
${\rm{\Delta }}_2^1$
-subset, and this is equivalent to the statement that there is a nonconstructible real.
(2)Every uncountable
${\rm{\Sigma }}_2^1$
-set has an uncountable
${\rm{\Delta }}_2^1$
-subset if and only if either every real is constructible or
$\omega _1^L$
is countable.
We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect
${\rm{\Pi }}_2^1$
-set with no nonempty
${\rm{\Pi }}_2^1$
-thin subset, strengthening a result of Harrington [4].
We analyze the models
$L[T_{2n} ]$
, where
$T_{2n}$
is a tree on
$\omega \times \kappa _{2n + 1}^1 $
projecting to a universal
${\rm{\Pi }}_{2n}^1 $
set of reals, for
$n > 1$
. Following Hjorth’s work on
$L[T_2 ]$
, we show that under
${\rm{Det}}\left( {{\rm{}}_{2n}^1 } \right)$
, the models
$L[T_{2n} ]$
are unique, that is they do not depend of the choice of the tree
$T_{2n}$
. This requires a generalization of the Kechris–Martin theorem to all pointclasses
${\rm{\Pi }}_{2n + 1}^1$
. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of
${\cal M}_{2n + 1}^\# $
. We then show that the GCH holds in
$L[T_{2n} ]$
, for every
$n < \omega $
, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.
Given a family
${\cal C}$
of infinite subsets of
${\Bbb N}$
, we study when there is a Borel function
$S:2^{\Bbb N} \to 2^{\Bbb N} $
such that for every infinite
$x \in 2^{\Bbb N} $
,
$S\left( x \right) \in {\Cal C}$
and
$S\left( x \right) \subseteq x$
. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an
$F_\sigma $
tall ideal on
${\Bbb N}$
without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a
${\bf{\Pi }}_2^1 $
tall ideal on
${\Bbb N}$
without a tall closed subset.
Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand.28 (1971), 124–128; Israel J. Math.13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo
$\text{ZF}+\text{DC}$
, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on
$\{0,1\}^{\mathbb{N}}$
has finite chromatic number.