Title: HOD Pair Capturing.
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Abstract:
Assume {\sf AD}. A {\em least branch hod pair} is a pair $(P,\Sigma)$
such that $P$ is a premouse constructed from a coherent
sequence of extenders together with a predicate for $\Sigma$,
and $\Sigma$ is an iteration strategy for $P$ having certain
regularity properties. Such pairs can be used to analyze HOD
fine structurally, provided that there are enough of them.
More precisely, one needs that every Suslin, co-Suslin set of reals
is Wadge reducible to the codeset of some lbr hod pair.
{\em HOD Pair Capturing} ({\sf HPC}) asserts that this is the case.
{\sf HPC} is the fundamental open question in the theory of HOD
in models of {\sf AD} which do not have iteration strategies for
mice with long extenders. We shall discuss what is known about it.
Title: On measures, P-measures and measure avoiding ultrafilters.
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Abstract: I will overview recent results concerning some special
points in $\beta\omega$ and some special measures on $\omega$. In
particular I will present the notion of measure avoiding ultrafilters
and I will discuss the problem of existence of P-measures in the Silver
model. This represents joint works with Jonathan Cancino and Adam Morawski
and with Artsiom Ranchinski.
Title: Games for chromatic numbers of analytic graphs.
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Abstract: We define games which characterize countable coloring numbers of
analytic graphs on Polish spaces. These games can provide simple
verification of the countable chromatic number of certain graphs. We
also get a simpler proof of a dichotomy originally proved by Adams and
Zapletal: if an analytic graph has an uncountable coloring number, then
it contains a certain subgraph.
Joint work with Jindrich Zapletal.
Title: Spectra and Definability.
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Abstract: Maximal almost disjoint families, maximal cofinitary groups and maximal independent families are among the combinatorial sets of reals, which are central to the study of the
set theoretic properties of the real line. In this talk, we will discuss recent developments regarding the possible cardinalities of such extremal sets of reals, as well as their definability properties.
Title: A quantifier elimination theorem for Weak König's Lemma with a negated induction axiom.
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Weak König's Lemma ($\mathrm{WKL}$) is a sentence of second-order arithmetic stating that every infinite binary tree has an infinite branch. WKL plays a significant role in the research programme called reverse mathematics, which attempts to characterize the strength of mathematical theorems by proving them equivalent (over a suitable base theory) to various set existence axioms.
$\mathrm{WKL}$ is known to be equivalent to basic theorems from many areas of mathematics, ranging from Peano's existence theorem for ODEs to the completeness theorem for first-order logic.
The default base theory used to prove equivalences in reverse mathematics, $\mathrm{RCA}_0$, is axiomatized by the comprehension scheme for $\Delta^0_1$-definable properties of natural numbers and by induction for $\Sigma^0_1$-definable properties. A fundamental fact about $\mathrm{WKL}$ (first proved by Harrington in the 1970s by a forcing argument) is that adding it to $\mathrm{RCA}_0$ results in a $\Pi^1_1$-conservative extension, in the sense that no new $\Pi^1_1$ statements become provable. $\mathrm{WKL}$ is neither the only nor the strongest $\Pi^1_2$ statement that is $\Pi^1_1$-conservative over $\mathrm{RCA}_0$.
It was shown in the 1980s that $\Pi^1_1$-conservativity of $\mathrm{WKL}$ still holds if we replace $\mathrm{RCA}_0$ by the alternative weaker base theory $\mathrm{RCA}^*_0$, which allows induction only for $\Delta^0_1$-definable as opposed to $\Sigma^0_1$-definable properties. However, we show that a new phenomenon emerges in models of $\mathrm{RCA}^*_0$ that are not models of $\mathrm{RCA}_0$ -- in other words, they contain a set relative to which $\Sigma^0_1$ induction \emph{fails}. We prove that in that setting $\mathrm{WKL}$ implies elimination of second-order quantifiers -- i.e., a collapse of the analytic hierarchy. As a consequence, $\mathrm{WKL}$ is the strongest $\Pi^1_2$ statement that is $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg \mathrm{RCA}_0$.
Our analysis of models of $\mathrm{RCA}^*_0 + \neg \mathrm{RCA}_0 + \mathrm{WKL}$ also provides a solution to a problem of Towsner about the computability-theoretic complexity of the conservativity relation, and it allows us to reduce the major open problem of characterizing the $\Pi^1_1$ consequences of Ramsey's theorem for pairs to the special case where the arithmetical part of the $\Pi^1_1$ statements is at most $\Pi^0_5$.
Joint work with Marta Fiori Carones, Tin Lok Wong, and Keita Yokoyama.
Title: Continuous logic and equivalence relations.
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Abstract: We will discuss two applications of infinitary continuous logic to Borel complexity of equivalence relations. We will characterize in model-theoretic terms essentially countable isomorphism relations on Borel classes of locally compact Polish metric structures. This gives a new proof of Kechris' theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. We will also show that isomorphism on such classes is always Borel reducible to graph isomorphism. This immediately answers a question of Gao and Kechris whether isometry of locally compact Polish metric spaces is reducible to graph isomorphism. The first result is joint work with Andreas Hallbäck and Todor Tsankov.
Title: Determinacy and generic absoluteness for the definable powerset of the universally Baire sets.
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Abstract: Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of the universally Baire sets $\Gamma^\infty$ and show that, after collapsing a large cardinal, $L(\mathcal{A}^\infty)$ is a model of determinacy and its theory cannot be changed by forcing. Our main technical tool is an iteration that realizes the universally Baire sets as the sets of reals in a derived model of some iterate of $V$, from a supercompact cardinal $\kappa$ and a proper class of Woodin cardinals. This is joint work with Grigor Sargsyan.
Title: Was Ulam right?
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Abstract: An Ulam matrix is one of the earliest gems of infinite
combinatorics. Around the same time of its discovery, another Polish
mathematician, Wacław Sierpiński was toying with pathological
consequences of the continuum hypothesis. We shall argue that one of
these consequences gives an improved Ulam matrix.
We shall then embark into a systematic study of these principles,
identifying a weakness in the definition of an Ulam matrix (and its
extension due to Hajnal) in view of its intended application: preventing
a kappa-complete proper ideal over an uncountable cardinal kappa from
being weakly kappa-saturated. It will be shown that a cardinal kappa
admits an Ulam-type matrix iff kappa admits a nontrivial C-sequence (a
concept arising from the theory of walks on ordinals).
This is joint work with Tanmay Inamdar.
Title: Universality properties of graph homomorphism: one construction to prove them all.
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Abstract: We show that the natural operation of connected sum for graphs can be used to prove at once most of the universality results from the literature concerning graph homomorphism. In doing so, we significantly improve many existing theorems and we also solve some natural open problems. Despite its simplicity, our technique unexpectedly leads to applications in quite diverse areas of mathematics, such as category theory, combinatorics, classical descriptive set theory, generalized descriptive set theory, model theory, and theoretical computer science.
Title: The definability of the non-stationary ideal.
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Title: Descriptive set theory in generalized Baire spaces.
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Title: Approachable Free Subsets and a question of Pereira.
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We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the pcf-conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:
Theorem. If ABSP holds for an omega sequence of regular cardinals with supremum \mu then there is an inner model with measurables below \mu but of arbitrarily large Mitchell order below \mu.
A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength. Their result went via the consistency of the non-existence of continuous tree-like scales; the result here is direct and avoids the use of PCF-like scales.
Title: On countably perfectly meager sets.
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We say that a subset A of a perfect Polish space X is countably perfectly meager in X, if for every sequence of perfect subsets {P_n : n \in N } of X, there exists an F_sigma-set F in X such that A is a subset of F and the intersection of F with P_n is meager in P_n for each n. This is an apparent strengthening not only of the classical notion of a perfectly meager set but also that of a universally meager set, i.e., a subset A of X which does not contain any injective Borel image of a non-meager set. But are the two classes of sets really different? We show that it is so if continuum is at most aleph_2 .
The talk is based on results from joint papers with Roman Pol and Tomasz Weiss:
R. Pol, P. Zakrzewski, Countably perfectly meager sets, J. Symbolic Logic 86(3) (2021), 1-17.
T. Weiss, P. Zakrzewski, Countably perfectly meager and countably perfectly null sets, arXiv:2304.07579 [math.LO].
Title: Around Mycielski and Eggleston theorems.
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Abstract: Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider strengthening of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of some types of trees with $A\subseteq B$.
Eggleston theorem says that every measurable set $X\subseteq [0,1]^2$ of positive measure contains a rectangle of the form $P\times Q$, where $P, Q$ are perfect sets and $Q$ has positive measure. We consider analogons and strengthening of this theorem in a similar manner as in Mycielski theorem.