## Zoltán Vidnyánszky: Borel chromatic numbers: basis and antibasis results

Talk held by Zoltán Vidnyánszky (KGRC) at the KGRC seminar on 2018-06-14.

Abstract: We give a full description of the existence of a homomorphism basis for Borel graphs of given Borel chromatic number. In particular, we show that there is a Borel graph with Borel chromatic number 3 that admits a homomorphism to any Borel graph of Borel chromatic number at least 3. We also discuss the relation of these results to Hedetniemi’s conjecture.

## Andrea Medini: Homogeneous spaces and Wadge theory

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2018-06-07.

Abstract: All spaces are assumed to be separable and metrizable. A space $X$ is homogeneous if for all $x,y\in X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$. A space $X$ is strongly homogeneous if all non-empty clopen subspaces of $X$ are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of $2^\omega$.

This is joint work with Raphaël Carroy and Sandra Müller.

## Borisha Kuzeljevic: P-ideal dichotomy and some versions of the Souslin Hypothesis

Talk held by Borisha Kuzeljevic (Czech Academy of Sciences, Prague) at the KGRC seminar on 2018-05-24.

Abstract: The talk will be about the relationship of PID with the statement that all Aronszajn trees are special. This is joint work with Stevo Todorcevic.

## Yair Hayut: Stationary reflection at $\aleph_{\omega+1}$

Talk held by Yair Hayut (Tel-Aviv University, Israel)
at the KGRC seminar on 2018-05-17.

Abstract: Stationary reflection is one of the basic prototypes of reflection phenomena,
and its failure is related to many counterexamples for compactness
properties (such as almost free non-free abelian groups,
and more). In 1982, Magidor showed that it is consistent, relative to infinitely many
supercomapct cardinals, that stationary reflections holds at $\aleph_{\omega + 1}$.
In this talk I’m going to present a new method for forcing stationary reflection
at $\aleph_{\omega+1}$, which allows to significantly reduce the upper bound for the consistency strength of the full stationary reflection at $\aleph_{\omega+1}$ (below a single partially supercompact cardinal).

This is a joint work with Spencer Unger.

## Vincenzo Dimonte: The sensitive issue of iterability

Talk held by Vincenzo Dimonte (University of Udine, Italy) at the KGRC seminar on 2018-05-03.

Abstract: In the momentous years when the community of set theorists was reaching the definite answer for the problem of the consistency of the Axiom of Determinacy, Martin wrote a small paper in the Proceedings of the International Congress of Mathematicians, 1978, in which he proved that the iterable version of I3, a very large cardinal, implied the determinacy of $\Pi^1_2$ sets of reals. Later it was proved that AD had much lower consistency, and iterable I3 fell into oblivion. In the last decade interest on I3 re-emerged, but iterable I3 is still elusive, and the small paper by Martin is not helpful, as it is terse and full of gaps. Even the definition of iterable I3 is not convincing. In this seminar we will bring back to life this abandoned hypothesis, clean it up to modern standards, and reveal the existence of a new hierarchy of axioms that was previously overlooked.

## Filippo Calderoni: The bi-embeddability relation for countable abelian groups

Talk held by Filippo Calderoni (Università di Torino, Italy and Politecnico di Torino, Italy) at the KGRC seminar on 2018-04-26.

Abstract: We analyze the Borel complexity of the bi‑embeddability relation for different classes of countable abelian groups. Most notably, we use the Ulm theory to prove that bi‑embeddability is incomparable with isomorphism in the case of p‑groups, and torsion groups. As I will explain, our result contrasts the arguable thesis that the bi‑embeddability relation on countable abelian p‑groups has strictly simpler complete invariants than isomorphism.

This is joint work with Simon Thomas.

## Diana Carolina Montoya: On some ideals associated with independent families

Talk held by Diana Carolina Montoya (KGRC) at the KGRC research seminar on 2018-04-19.

Title: On some ideals associated with independent families

Abstract. The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.

In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah’s proof of the consistency of $\mathfrak i\lt \mathfrak u$ (here $\mathfrak i$ and $\mathfrak u$ are  the independence and ultrafilter numbers respectively). Additionally, we study the relationship  between these two ideals for an arbitrary independent family $A$, and define a class of maximal  independent families — which we call densely independent — for which the ideals mentioned above  coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for  such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

This is joint work with Vera Fischer.

## Victoria Gitman: Virtual large cardinal principles

Research Seminar, Kurt Gödel Research Center, Thursday, April 12

Speaker: Victoria Gitman, (Graduate Center, City University of New York (CUNY), USA)

Abstract: Given a set-theoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some first-order structures, we say that $\mathcal P$ holds virtually if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$-extendible, $n$-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with $V=L$. Sitting atop the hierarchy are virtual versions of inconsistent large cardinal principles such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$,are also natural examples of virtual large cardinals. With Bagaria, Hamkins and Schindler, we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle. Time permitting, I will give an overview of Woodin’s new results on virtual large cardinals in cardinal preserving extensions.we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle.

# Set theory today: A conference in honor of Georg Cantor

## Confirmed speakers:

• Omer Ben Neria, UCLA.
• Jörg Brendle, Kobe University.
• David Chodounský, Institute of Mathematics of the Czech Academy of Sciences.
• James Cummings, Carnegie Mellon University.
• Oswaldo Guzmán, York University.
• Radek Honzik, Charles University in Prague.
• Yurii Khomskii, University of Hamburg.
• Paul Larson, Miami University.
• Diego Mejía, Shizuoka University.
• Julien Melleray, University of Lyon.
• Heike Mildenberger, University of Freiburg.
• Luca Motto Ros, University of Torino.
• Christian Rosendal, University of Illinois at Chicago.
• Grigor Sargsyan, Rutgers University.
• Asger Törnquist, University of Copenhagen.
• Todor Tsankov, University of Paris Diderot.
• Matteo Viale, University of Torino.
• Jindřich Zapletal, University of Florida.

For a list of confirmed participants please see

## Organizing/scientific committee

• Vera Fischer
• Sy-David Friedman
• Benjamin Miller

## Local organizing committee

• Diana Carolina Montoya
• Daniel Soukup

## Monroe Eskew: Local saturation of the nonstationary ideals

Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-03-22.

Abstract: It is consistent relative to a huge cardinal that for all successor cardinals $\kappa$, there is a stationary $S \subseteq \kappa$ such that the nonstationary ideal on $\kappa$ restricted to $S$ is $\kappa^+$-saturated. We will describe the construction of the model, focusing how to get this property on all $\aleph_n$ simultaneously. Time permitting, we will also briefly discuss the Prikry-type forcing that extends this up to $\aleph_{\omega+1}$.