Narrow systems and the singular cardinals hypothesis
(RIMS Set Theory Workshop, Kyoto, Japan, October 2022)
(Slides)
Two-cardinal tree properties, and their effect on cardinal arithmetic, have been
a central topic of study in combinatorial set theory over the last fifteen years. Notably, a pair
of results due to Viale and Krueger, respectively, shows that the strongest of these principles,
\(\mathsf{ISP}(\kappa)\), implies that the singular cardinals hypothesis holds above \(\kappa\).
This raises the natural question of whether the same conclusion follows from the weaker
\(\mathsf{ITP}(\kappa)\) or the strong tree property at \(\kappa\). We address this question by introducing
the notion of a generalized narrow system and the generalized narrow system property.
This generalized narrow system property holds in all known models of the strong tree property
and is often implicitly used in the verification of the strong tree property. We will show that
the generalized narrow system property at a cardinal \(\kappa\) implies the singular cardinals
hypothesis above \(\kappa\). We also present a consistency result showing that the generalized
narrow system property can hold globally.
Adding many Cohen reals (European Set Theory Conference 2022, Turin,
Italy, September 2022)
(Slides)
We survey some recent work exploring the effects of
adding a large number of Cohen reals to
models of ZFC. We begin by discussing a few purely combinatorial results
indicating that, after adding many Cohen reals, a number of nontrivial positive partition relations
necessarily hold at the level of the continuum. In all cases, these partition relations are known to be
inconsistent with the continuum being small. We then turn to questions about the preservation of
compactness principles under Cohen forcing, focusing in particular on the Guessing Model Principle
and its relatives. Finally, we discuss the effect of adding Cohen reals on the vanishing of certain
higher derived inverse limits, addressing questions arising from homological algebra and algebraic
topology. Throughout, we will attempt to highlight the common themes and techniques that
underlie and connect these results. The talk will include joint work with Jeffrey Bergfalk, Radek Honzik,
Michael Hrušák, Šárka Stejskalová, and Andy Zucker.
Some applications of covering matrices
(SETTOP 2022, Novi Sad, Serbia, August 2022)
(Slides)
Covering matrices were introduced by Matteo Viale in
his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing
Axiom, and they are a useful tool when attempting to propagate certain
combinatorial statements through singular cardinals. In this talk, we present
three recent applications of covering matrices. The first concerns the effect
of the Guessing Model Principle on cardinal arithmetic. In the second, building
on work of Chen-Mertens and Szeptycki, we show that the failure of SCH entails
the existence of a Fréchet, \(\alpha_1\)-space whose \(G_\delta\)-modification
has large tightness. In the third, we prove a result about the effect of
approachability on compactness for coloring numbers of graphs. This talk
contains joint work with Assaf Rinot and with Šárka Stejskalová.
Two-cardinal combinatorics, guessing models, and cardinal arithmetic
(Advances in Set Theory 2022, Hebrew University of Jerusalem, July 2022)
(
Slides)
Two-cardinal tree properties were first studied by
Jech and Magidor in the 1970s and were used to characterize strongly compact
and supercompact cardinals. In the 2000s, Weiss formulated the two-cardinal
tree properties (I)TP and (I)SP, which capture many of the combinatorial
consequences of strong compactness and supercompactness but also consistently
hold at small cardinals. Subsequently, Viale and Weiss showed that at \(\omega_2\),
ISP, the strongest of these principles, is equivalent to the Guessing Model Property
(GMP), which has proven to be a very fruitful set theoretic hypothesis. In this
talk, we will discuss some recent results concerning these principles and their
variants. We will begin by introducing weakenings of the GMP that are equivalent to
SP or even further weakenings thereof but still have considerable combinatorial
consequences, for instance implying global failures of square. Then, motivated
by the question as to whether (I)TP at a cardinal \(\kappa\) implies the Singular Cardinals Hypothesis
(SCH) above \(\kappa\), we will discuss the consequences of various related
two-cardinal combinatorial principles, including generalized narrow system properties
and the non-existence of certain strongly unbounded subadditive functions, on cardinal arithmetic,
particularly on SCH and Shelah's Strong Hypothesis. This is joint work with Šárka Stejskalová.
A Galvin-Hajnal theorem for generalized cardinal characteristics
(Bar-Ilan University Set Theory Seminar, June 2022)
We prove that a variety of generalized cardinal characteristics,
including meeting numbers, the reaping number, and the dominating number, satisfy an
analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular
cardinals of uncountable cofinality. In the first talk, we will introduce some fundamental
PCF-theoretic facts due to Shelah and Jech that will be used in our proof, and in the
second talk we will prove our generalized Galvin-Hajnal theorem.
Nontrivial coherent families of functions (Series of tutorials at the
Winter School in Abstract Analysis, Hejnice, Czech Republic, February 2022)
(Slides I,
Slides II,
Slides III)
In this tutorial, we will survey the history of
and recent developments in the set theoretic study of nontrivial coherent
families of functions, focusing in particular on families indexed by the space
\({^\omega}\omega\). Such families originally arose out of homological considerations
(they can be seen as witnesses to the nontriviality of certain derived inverse limits,
or as witnesses to the nonadditivity of strong homology), but they can also be thought
of as purely set theoretic objects of interest in their own right. In the first half of
the tutorial, we will cover work from the late 1980s and early 1990s connecting the
existence of 1-dimensional nontrivial coherent families with topics such as cardinal
characteristics of the continuum and the Open Coloring Axiom. In the second half,
we will cover some recent results about higher-dimensional nontrivial coherent
families. The focus of the tutorial will be on the set theoretic aspects of the topic,
but we will also touch on its origins in and continued applications to homological algebra.
Strong tree properties, the Kurepa hypothesis, and the continuum
(Mathematical Logic Seminar of the Institute for Research in Fundamental
Sciences, Tehran, November 2021)
In 2010, Christoph Weiss isolated a family of
strong tree properties that can be used to characterize strongly compact
or supercompact cardinals but can also consistently hold at smaller cardinals.
For example, the Proper Forcing Axiom implies that even the strongest of these
tree properties, ISP, holds at \(\omega_2\). In this talk, we will investigate
a variety of interactions between these strong tree properties, the existence of
(weak) Kurepa trees, and the behavior of the continuum function. Along the way,
we will also encounter a variant of Silver's celebrated theorem on cardinal
arithmetic dealing with a certain cardinal characteristic of singular cardinals.
This is joint work with Šárka Stejskalová.
Variations on a theorem of Silver (Logic Colloquium of the
Kurt Gödel Research Center, October 2021)
Shortly after the advent of forcing in the 1960s, Easton proved that,
modulo some trivial constraints concerning monotonicity and cofinality, the axioms
of set theory place no restrictions on the behavior of exponentiation at regular cardinals. In a
surprising turn of events, this turned out not to be the case for singular cardinals,
and the last half-century has seen a procession of deep results uncovering nontrivial constraints on
exponentiation at singular cardinals. One of the first of these results was Silver's theorem,
which in essence states that if \(\lambda\) is a singular cardinal of uncountable cofinality
and there are "many" singular cardinals \(\kappa < \lambda\) such that \(2^\kappa = \kappa^+\),
then it must also be the case that \(2^\lambda = \lambda^+\). In particular, if the Singular Cardinals
Hypothesis fails, then it must fail first at a singular cardinal of countable cofinality. We will discuss
this seminal theorem and a number of variations thereon, and we will end by sketching a proof of
a version of Silver's theorem pertaining to certain generalized cardinal characteristics.
Strongly unbounded subadditive colorings (Set Theory Seminar of the
Kurt Gödel Research Center, October 2021)
Given infinite cardinals \(\kappa\) and \(\theta\), functions of the form
\(c:[\kappa]^2 \rightarrow \theta\) exhibiting certain unboundedness properties
provide a strong counterexample to the generalization of Ramsey's theorem to \(\kappa\) and have seen a
wide variety of applications. In this talk, we will discuss the existence of such strongly
unbounded colorings, focusing in particular on colorings with subadditivity properties.
We will then present some applications to general topology. In particular, building on
work of Chen-Mertens and Szeptycki, we will prove that the failure of the Singular
Cardinals Hypothesis implies the existence of a Fréchet, \(\alpha_1\)-space whose
\(G_\delta\)-modification has large tightness. This is joint work with Assaf Rinot.
Strongly unbounded colorings (Kobe Set Theory Workshop on the occasion of
Sakaé Fuchino's retirement, Kobe, Japan, March 2021)
(Video)
(Slides)
For infinite cardinals \(\theta < \kappa\), colorings
of the form \(f:[\kappa]^2 \rightarrow \theta\) that exhibit certain strong
unboundedness properties have seen a wide variety of applications. In this
talk, we will discuss some results about the existence of such strongly unbounded
colorings as well as strongly unbounded colorings that satisfy additional
closure or subadditivity requirements. We will then present some recent
applications of these colorings to questions regarding the infinite productivity
of strong chain conditions and the tightness of topological spaces. This is
joint work with Assaf Rinot.
Nontrivial coherent families of functions (South Eastern Logic Symposium,
University of Florida, February 2021)
In the 1980s, considerations in homological algebra
gave rise to combinatorial set theoretic questions about nontrivial coherent
families of functions indexed by elements of the Baire space. The existence
of such families turned out to be intimately connected to cardinal characteristics
of the continuum and the Open Coloring Axiom. Similar homological considerations
naturally give rise to higher-dimensional analogues of these nontrivial coherent
families of functions. We will begin the talk by introducing these families and
indicating how they arise from algebraic questions about the derived functors
of the inverse limit functor. We will then sketch a proof of the fact that,
in the forcing extension obtained by adding \(\beth_\omega\)-many Cohen reals,
for every \(n \geq 1\), every \(n\)-dimensional coherent family of functions indexed
by \(({^\omega}\omega)^n\) is trivial. We end with a broader discussion of some
of the implications of this result and its techniques and of directions for
further research. This is joint work with Jeffrey Bergfalk and Michael Hrusak.
Pseudo-Prikry sequences (Prikry Forcing Online, University of East
Anglia, December 2020)
(Slides)
Prikry-type forcing notions have been a
uniquely effective tool for proving consistency results about singular
cardinal combinatorics. In this talk, we will survey a number of results
giving some indication as to why this is the case. In particular, we will
show that, in a wide variety of circumstances in which \(V \subseteq W\)
are models of set theory such that there is at least one regular cardinal
of \(V\) that is singular in \(W\) (or such that certain PCF-theoretic
relations hold between the models), there provably exists an object in
\(W\) that approximates a generic object over \(V\) for some Prikry-type
forcing notion. We will begin the talk by discussing the earliest work
in this direction, done by Gitik and by Džamonja and Shelah in the
1990s, and then turn to more recent work by Magidor and Sinapova, Gitik,
and the speaker.
Higher dimensional Delta-systems (Cornell University Logic Seminar,
December 2020.)
(Slides)
The classical \(\Delta\)-system lemma is
one of the foundational results of combinatorial set theory and is
an important tool in many forcing arguments. \(\Delta\)-systems can
be seen as inherently one-dimensional objects, though, so arguments
about higher-dimensional phenomena often call for higher-dimensional
generalizations of the classical \(\Delta\)-system lemma. Such
generalizations were first developed and applied by Todorcevic and Shelah
in the 1980s, and they have seen increased use in recent years. In this
talk, we will present a particular definition of ``higher-dimensional
\(\Delta\)-system", isolate optimal hypotheses under which
such generalized \(\Delta\)-systems necessarily exist, and present some
applications to questions arising from Ramsey theory and homological algebra.
Highly connected Ramsey theory. (RIMS Set
Theory Workshop, Kyoto, Japan, November 2020.)
(Slides)
In recent work, Bergfalk, Hrusak, and
Shelah introduce some natural nontrivial weakenings of the classical
partition relation \(\mu \rightarrow (\mu)^2_\kappa\) that can consistently
hold for uncountable but non-weakly compact cardinals \(\mu\). Roughly
speaking, these weakenings replace the assertion of the existence of
large monochromatic complete subgraphs in the classical partition
relation with the assertion of the existence of large highly connected
monochromatic subgraphs. We present some recent work on this topic,
including some results about the influence of PFA and square principles
on these partition relations. Our main result is a sharp consistency
result indicating that a square-bracket version of this partition
relation can consistently hold at the continuum, and that this is in
fact equiconsistent with the existence of a weakly compact cardinal.
Time permitting, we will discuss some potential higher-dimensional
generalizations of these results.
Set theoretic compactness and homological algebra. (VCU Analysis,
Logic, and Physics Seminar, August 2020.)
The study of compactness phenomena, or the extent to which structures' global behavior
reflects the behavior of their small substructures, has been the focus of a great deal
of set theoretic work since at least the mid-twentieth century, and this work has had applications
not just to set theory and logic but to fields throughout mathematics. In this talk,
we will discuss some set theoretic contributions to questions arising from homological
algebra, focusing in particular on the study of the derived limits of certain inverse
systems of abelian groups. We will show how the vanishing of these derived limits can be
translated to a purely combinatorial statement about the existence of nontrivial
coherent families of functions, and we will see how set theoretic techniques can shed light
on the circumstances under which such families do or do not exist. The talk will include
some joint work with Jeffrey Bergfalk and Michael Hrusak. No specialized knowledge
of either set theory or homological algebra will be assumed from the audience.
Finite subgraphs of uncountable graphs. (Carnegie Mellon Mathematical
Logic Seminar, April 2020.)
(Slides)
A prominent line of inquiry in the study of infinite graphs
concerns the extent to which global properties of an infinite graph are determined by
the structure of the set of its finite subgraphs. In this talk, we consider a question
in this direction concerning chromatic numbers of graphs. By a classic result of
De Bruijn and Erdős, if a graph \(G\) has infinite chromatic number, then it has
finite subgraphs of every finite chromatic number. We prove, however, that even for a
graph with uncountable chromatic number, the chromatic numbers of its finite subgraphs
can grow arbitrarily slowly. In particular, we prove that, for every function
\(f:\mathbb{N} \rightarrow \mathbb{N}\), there is a graph \(G\) of uncountable chromatic
number such that, for every natural number \(k \geq 3\) and every subgraph \(H\) of \(G\) with at most
\(f(k)\) vertices, the chromatic number of \(H\) is at most \(k\). This answers a question of
Erdős, Hajnal, and Szemerédi from 1982.
Finite subgraphs of uncountable graphs. (Rutgers Logic Seminar,
February 2020.)
A major line of inquiry in infinite graph theory
has been around the extent to which one can determine the global structure
of an infinite graph simply by looking at its finite subgraphs.
An early result in this direction, which is a consequence of the De
Bruijn-Erdős compactness theorem, shows that if \(G\) is a graph with
infinite chromatic number, then \(G\) contains finite subgraphs of every
possible finite chromatic number. One can then define a function
\(f_G:\mathbb{N} \rightarrow \mathbb{N}\) by letting \(f_G(k)\) be the least
number of vertices in a subgraph of \(G\) with chromatic number \(k\).
\(f_G\) is an increasing function, and a natural question to ask is how
quickly \(f_G\) can grow. Results of Erdős show that, for every function
\(f:\mathbb{N} \rightarrow \mathbb{N}\), there is a graph \(G\) with
countably infinite chromatic number such that \(f_G\) grows faster
than \(f\). In 1982, Erdős, Hajnal, and Szemerédi asked if the analogous
statement is true if we moreover require that \(G\) have uncountable
chromatic number. We will answer this question and more broadly discuss
other problems involving finite subgraphs of uncountable graphs.
Set theoretic compactness and higher derived limits. (CUNY Set Theory
Seminar, January 2020.)
Issues of set theoretic compactness frequently
arise when considering questions from homological algebra about derived functors. In particular,
the non-vanishing of such derived functors is often witnessed by a concrete
combinatorial instance of set theoretic incompactness. In this talk, we
will discuss some recent results about the derived functors of the inverse
limit functor. We will focus on a specific inverse system of abelian groups,
\(\mathbf{A}\), that arose in Mardešić and Prasolov's work on the
additivity of strong homology and has since arisen independently in a
number of contexts. Our main result states that, relative to the consistency
of a weakly compact cardinal, it is consistent that the \(n\)-th derived
limits \(\lim^n \mathbf{A}\) vanish simultaneously for all \(n \geq 1\). We
will sketch a proof of this fact and then discuss the extent to which
certain generalizations of this result can hold.
This is joint work with Jeffrey Bergfalk.
Set theoretic compactness and higher derived limits. (Toronto Set
Theory Seminar, November 2019.)
Issues of set theoretic compactness frequently
arise when considering questions about derived functors. In particular,
the non-vanishing of such derived functors is often witnessed by a concrete
combinatorial instance of set theoretic incompactness. In this talk, we
will discuss some recent results about the derived functors of the inverse
limit functor. We will focus on a specific inverse system of abelian groups,
\(\mathbf{A}\), that arose in Mardešić and Prasolov's work on the
additivity of strong homology and has since arisen independently in a
number of contexts. Our main result states that, relative to the consistency
of a weakly compact cardinal, it is consistent that the \(n\)-th derived
limits \(\lim^n \mathbf{A}\) vanish simultaneously for all \(n \geq 1\). We
will sketch a proof of this fact and then discuss the extent to which
certain generalizations of this result can hold.
This is joint work with Jeffrey Bergfalk.
Finite subgraphs of uncountable graphs. (VCU Discrete Mathematics
Seminar, October 2019.)
A major line of inquiry in infinite graph theory
has been around the extent to which one can determine the global structure
of an infinite graph simply by looking at its finite subgraphs.
An early result in this direction, which is a consequence of the De
Bruijn-Erdős compactness theorem, shows that if \(G\) is a graph with
infinite chromatic number, then \(G\) contains finite subgraphs of every
possible finite chromatic number. One can then define a function
\(f_G:\mathbb{N} \rightarrow \mathbb{N}\) by letting \(f_G(k)\) be the least
number of vertices in a subgraph of \(G\) with chromatic number \(k\).
\(f_G\) is an increasing function, and a natural question to ask is how
quickly \(f_G\) can grow. Results of Erdős show that, for every function
\(f:\mathbb{N} \rightarrow \mathbb{N}\), there is a graph \(G\) with
countably infinite chromatic number such that \(f_G\) grows faster
than \(f\). In 1982, Erdős, Hajnal, and Szemerédi asked if the analogous
statement is true if we moreover require that \(G\) have uncountable
chromatic number. We will answer this question and more broadly discuss
other problems involving finite subgraphs of uncountable graphs.
Simultaneously vanishing higher derived limits. (Set Theory of the Reals
BIRS-CMO Workshop, August 2019.)
(Slides)
In 1988, Mardešić and Prasolov
identified an inverse system \(\mathbf{A}\) such that the additivity of
strong homology on any reasonably nice class of spaces entails that
\(\lim^n \mathbf{A} = 0\) for all \(n > 0\). The vanishing of
\(\lim^n \mathbf{A}\) can be translated into a purely set theoretic
statement asserting the non-existence of certain non-trivial coherent families
of functions indexed by \(n\)-tuples from the Baire space. Mardešić
and Prasolov proved that CH implies \(\lim^1 \mathbf{A} \neq 0\), and, soon
afterward, Dow, Simon, and Vaughan proved that PFA implies \(\lim^1 \mathbf{A}
= 0\). Further significant set theoretic work on \(\lim^1 \mathbf{A}\) has
been done over the years, but \(\lim^n \mathbf{A}\) for \(n > 1\) has
remained more mysterious, and it even remained open whether
\(\lim^1 \mathbf{A}\) and \(\lim^2 \mathbf{A}\) could consistently vanish
simultaneously. We answer this, modulo a large cardinal assumption, by showing
that, after forcing with a finite support iteration adding a measurable number
of Hechler reals, \(\lim^n \mathbf{A} = 0\) for all \(n > 0\). This is
joint work with Jeffrey Bergfalk.
Finite subgraphs of uncountable graphs. (UNAM-Morelia Set Theory and Topology Seminar,
May 2019.)
The De Bruijn-Erdős Compactness Theorem states
that, if \(G\) is a graph, \(k\) is a natural number, and every finite subgraph
of \(G\) has chromatic number at most \(k\), then \(G\) has chromatic number
at most \(k\) as well. As a result, if \(G\) has infinite chromatic number,
then one can define a function \(f_G: \mathbb{N} \rightarrow \mathbb{N}\)
by letting \(f_G(k)\) be the least number of vertices in a \(k\)-chromatic
subgraph of \(G\). We prove that, for every function \(f: \mathbb{N} \rightarrow
\mathbb{N}\), there is a graph \(G\) with uncountable chromatic number such
that \(f_G\) grows faster than \(f\). This answers a question of
Erdős, Hajnal, and Szemerédi from 1982. Time permitting, we will
discuss connections between our proof and various diamond and club-guessing
principles.
Chromatic numbers of finite subgraphs. (Research Seminar of the Kurt Gödel Research Center,
March 2019.)
(Video)
By the De Bruijn-Erdős Compactness Theorem, if a graph \(G\) has infinite chromatic
number, then it has finite subgraphs of arbitrarily large finite chromatic number. We can therefore
define an increasing function \(f_G: \omega \rightarrow \omega\) by letting \(f_G(n)\) be the least number of
vertices in a subgraph of \(G\) with chromatic number \(n\). We will show in ZFC that, for every
function \(f:\omega \rightarrow \omega\), there is a graph \(G\) with chromatic number \(\aleph_1\)
such that \(f_G\) grows faster than \(f\). This answers a question of Erdős, Hajnal, and
Szemerédi. Time permitting, we will discuss connections between our proof and various
diamond and club-guessing principles.
The
C-sequence number. (CUNY Set Theory Seminar, March 2019.)
The C-sequence number of an uncountable regular cardinal \(\kappa\) is a cardinal invariant
that provides a measure of the amount of compactness that holds at \(\kappa\). We will begin this talk by
introducing the C-sequence number and proving some of its basic properties, linking it to familiar
notions including large cardinals and square principles. We will then outline a number of consistency
results regarding the C-sequence number at inaccessible cardinals and successors of singular cardinals.
We will end by exploring how the C-sequence number interacts with the existence of complicated colorings
and the infinite productivity of the \(\kappa\)-Knaster condition. This is joint work with Assaf Rinot.
Uncountable triangle-free graphs. (VCU Discrete Mathematics Seminar, December 2018.)
When one first encounters graph coloring, it quickly
becomes evident that the existence of many triangles (or, more generally, small cycles)
in a graph poses an obstacle to the graph having a small chromatic number. One might
then ask the question as to whether they pose the only obstacle: can a graph with
no triangles (or no cycles of length less than 5, etc.) have arbitrarily large
chromatic number? Of course, this question was answered in the relatively early
days of graph theory, and classical graph constructions yield finite graphs of
arbitrarily large girth and chromatic number. By taking disjoint unions of
such graphs, we can push these results into the world of the countably infinite,
producing countably infinite graphs of arbitrarily large girth whose chromatic numbers are infinite.
In this talk, we will consider some questions regarding generalizations of
these results to the realm of the uncountably infinite. We will begin by
noting a somewhat surprising way in which the results about finite and countable
graphs cannot generalize to the uncountable case. We will then view some
selections from a menagerie of uncountable graph curiosities, including a graph
which is cycle-free and yet, in some models of set theory (without the axiom of
choice), has uncountable chromatic number. We will end by outlining
a construction which involves arguments both combinatorial and set-theoretic
in nature and that yields, for every uncountable cardinal \(\kappa\), graphs of
size and chromatic number \(\kappa\) of arbitrarily large odd girth.
Compactness and incompactness in set theory. (VCU Analysis, Logic, Physics Seminar, August 2018.)
A statement of compactness, roughly speaking,
asserts that, if all "small" substructures of a given structure have a certain
property, then the entire structure has that property as well. A number of
seminal mathematical results from the early-to-mid twentieth century, among
them König's Infinity Lemma, Ramsey's Theorem, the compactness theorem
for first-order logic, and the De Bruijn-Erdős Theorem, can together be
seen as saying that the smallest infinite cardinal exhibits a high degree of
compactness. Since then, a great amount of work in set theory has been devoted
to understanding the extent to which similar compactness phenomena can occur
at uncountable cardinals. In this talk, we will give a survey of some highlights
of this work, touching on algebra, topology, and combinatorial set theory,
and culminating in some recent results involving the chromatic and coloring
numbers of uncountable graphs.
Unbounded colorings and the C-sequence number. (SETTOP 2018. Novi Sad, Serbia, July 2018.)
(Slides)
Motivated by questions about the infinite productivity
of strong chain conditions, we introduce and analyze a coloring principle
asserting the existence of certain strongly unbounded functions. We use this
principle to show, for instance, that the \(\kappa\)-Knaster property is not
infinitely productive for any successor cardinal \(\kappa\). We also introduce a
cardinal invariant, the C-sequence number, that is deeply connected to
our coloring principle and can be seen as a way of measuring the compactness
of an uncountable cardinal. We then present a number of ZFC theorems and
independence results concerning the C-sequence number and linking it to various
large cardinal notions. This is joint work with Assaf Rinot.
Unbounded functions and infinite productivity of the Knaster property.
(Set Theory, Model Theory and Applications, Eilat, April 2018.)
(Slides)
Motivated by questions about the circumstances
under which the \(\kappa\)-Knaster property can be infinitely productive, we study
principles asserting the existence of functions exhibiting certain strong
unboundedness properties. We derive such functions from failures of stationary
reflection, and we discuss applications to the infinite productivity of the
\(\kappa\)-Knaster property and to the topological question of the tightness of the
square of the sequential fan. This is joint work with Assaf Rinot.
Squares, ascent paths, and chain conditions.
(JMM, AMS-ASL Special Session on Set Theory, Logic, and Ramsey Theory, San Diego, January 2018.)
(Slides)
Using a variety of square principles,
we obtain results on the consistency strengths of the non-existence of
\(\kappa\)-Aronszajn trees with narrow ascent paths and of the infinite productivity
of strong \(\kappa\)-chain conditions. In particular, we show that, if \(\kappa\) is an
uncountable regular cardinal that is not weakly compact in L, then:
1. for every \(\lambda < \kappa\), there is a \(\kappa\)-Aronszajn tree with a \(\lambda\)-ascent path;
2. there is a \(\kappa\)-Knaster poset \(\mathbb{P}\) such that \(\mathbb{P}^\omega\)
does not have the \(\kappa\)-chain condition;
3. there is a \(\kappa\)-Knaster poset that is not \(\kappa\)-stationarily layered.
This answers questions of Cox and Lücke and consists of joint work with Philipp Lücke.
Reflections on graph coloring. (MAMLS Logic Friday, CUNY Graduate Center, October 2017.)
(Slides)
In 1951, de Bruijn and Erdős published a compactness
theorem for graphs with finite chromatic number, proving that, if \(G\) is a graph,
\(k\) is a natural number, and all finite subgraphs of \(G\) have chromatic number at most \(k\),
then \(G\) has chromatic number at most \(k\). Since then, infinitary generalizations of
this theorem, for the chromatic number as well as the coloring number of graphs,
have attracted much attention. In this talk, we will briefly review some of the
historical highlights in this area and then present some new work. These results
show that the coloring number can exhibit only a limited amount of incompactness,
while large amounts of incompactness for the chromatic number are implied by
relatively weak hypotheses. This indicates that the coloring number and chromatic
number behave quite differently with respect to compactness and illustrates the
difficulty involved in obtaining infinitary analogues of the de Bruijn-Erdős
result at infinite, accessible cardinals. This is joint work with Assaf Rinot.
A forcing axiom deciding the generalized Souslin Hypothesis.
(Carnegie Mellon University Logic Seminar, October 2017.)
Given a regular, uncountable cardinal \(\kappa\), it is
often desirable to be able to construct objects of size \(\kappa^+\) using approximations
of size less than \(\kappa\). Historically, such constructions have often been carried
out with the help of a \((\kappa, 1)\)-morass and/or a \(\diamondsuit(\kappa)\)-sequence.
We present a framework for carrying out such constructions using \(\diamondsuit(\kappa)\)
and a weakening of Jensen's principle \(\square_\kappa\). Our framework takes the
form of a forcing axiom, \(\mathrm{SDFA}(\mathcal{P}_\kappa)\). We show that
\(\mathrm{SDFA}(\mathcal{P}_\kappa)\) follows from
the conjunction of \(\diamondsuit(\kappa)\) and our weakening of \(\square_\kappa\) and, if
\(\kappa\) is the successor of an uncountable cardinal, that
\(\mathrm{SDFA}(\mathcal{P}_\kappa)\) is in fact
equivalent to this conjunction. We also show that, for an infinite cardinal \(\lambda\),
\(\mathrm{SDFA}(\mathcal{P}_{\lambda^+})\) implies the existence of a
\(\lambda^+\)-complete \(\lambda^{++}\)-Souslin tree.
This implies that, if \(\lambda\) is an uncountable cardinal, \(2^\lambda = \lambda^+\), and
Souslin's Hypothesis holds at \(\lambda^{++}\), then \(\lambda^{++}\) is a Mahlo cardinal in L,
improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.
A forcing axiom deciding the generalized Souslin Hypothesis.
(Miami University Logic Seminar, September 2017.)
Given a regular, uncountable cardinal \(\kappa\), we isolate a
forcing axiom, \(\mathrm{SDFA}(\mathcal{P}_\kappa)\), that provides a framework for constructing objects
of size \(\kappa^+\) using approximations of size \(<\kappa\). We show that, for uncountable
successor \(\kappa\), \(\mathrm{SDFA}(\mathcal{P}_\kappa)\) is equivalent to the conjunction of \(2^{ < \kappa} =
\kappa\) and \(\square^B_\kappa\), which is a weakening of Jensen's \(\square_\kappa\). In our
main application, we show that, for an infinite cardinal \(\lambda\), \(\mathrm{SDFA}(\mathcal{P}_{\lambda^+})\)
implies the existence of a \(\lambda^+\)-closed \(\lambda^{++}\)-Souslin tree. This yields a
corollary stating that, if \(\lambda\) is an uncountable cardinal, \(2^\lambda = \lambda^+\),
and Souslin's Hypothesis holds at \(\lambda^{++}\), then \(\lambda^{++}\) is Mahlo in L,
improving upon a result of Shelah and Stanley. This is joint work with Assaf Rinot.
Constructions from squares and diamonds. (6th European Set Theory Conference, Budapest,
July 2017.)
(Slides)
In 1982, Shelah and Stanley proved that, if \(\kappa\) is a regular,
infinite cardinal, \(2^\kappa = \kappa^+\), and there is a \((\kappa^+, 1)\)-morass, then there
is a \(\kappa^{++}\)-super-Souslin tree, which is a type of normal \(\kappa^{++}\)-tree that
necessarily has a \(\kappa^{++}\)-Souslin subtree and continues to do so in any
outer model in which \(\kappa^{++}\) is preserved and no new subsets of \(\kappa\) are
present. This result establishes a lower bound of an inaccessible cardinal
for the consistency strength of the conjunction of \(2^\kappa = \kappa^+\) and
Souslin's Hypothesis at \(\kappa^{++}\). In this talk, we will present a method
for constructing objects of size \(\lambda^+\) from \(\square_\lambda + \diamondsuit_\lambda\),
where \(\lambda\) is a regular, uncountable cardinal. As an application, we will
use \(\square_{\kappa^+} + \diamondsuit_{\kappa^+}\) to construct a \(\kappa^{++}\)-super-Souslin
tree. For uncountable \(\kappa\), this increases Shelah and Stanley's lower bound
from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.
Constructions from square and diamond, with an application to super-Souslin trees.
(Oberseminar mathematische Logik, University of Bonn, May 2017.)
In 1982, Shelah and Stanley proved that, if \(\kappa\) is a regular,
infinite cardinal, \(2^\kappa = \kappa^+\), and there is a \((\kappa^+, 1)\)-morass, then there
is a \(\kappa^{++}\)-super-Souslin tree, which is a type of normal \(\kappa^{++}\)-tree that
necessarily has a \(\kappa^{++}\)-Souslin subtree and continues to do so in any
outer model in which \(\kappa^{++}\) is preserved and no new subsets of \(\kappa\) are
present. This result establishes a lower bound of an inaccessible cardinal
for the consistency strength of the conjunction of \(2^\kappa = \kappa^+\) and
Souslin's Hypothesis at \(\kappa^{++}\). In this talk, we will present a method
for constructing objects of size \(\lambda^+\) from \(\square_\lambda + \diamondsuit_\lambda\),
where \(\lambda\) is a regular, uncountable cardinal. As an application, we will
use \(\square_{\kappa^+} + \diamondsuit_{\kappa^+}\) to construct a \(\kappa^{++}\)-super-Souslin
tree. For uncountable \(\kappa\), this increases Shelah and Stanley's lower bound
from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.
Reflections on the coloring and chromatic numbers. (University of Helsinki Logic Seminar, April 2017.)
Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century, when de Bruijn and Erdős published their compactness theorem for finite chromatic numbers. In this talk, we will briefly review some of the highlights in this area and then present new results indicating, firstly, that the coloring number can only exhibit a limited amount of incompactness, and, secondly, that large amounts of incompactness for the chromatic number are compatible with strong compactness statements, including compactness for the coloring number. This indicates that the chromatic and coloring numbers behave quite differently with respect to compactness. This is joint work with Assaf Rinot.
Partition relations and generalized scattered orders. (Bar-Ilan University Set Theory Seminar, March 2017.)
The class of scattered linear orders, isolated by Hausdorff,
plays a prominent role in the study of general linear orders. In 2006, Dzamonja and
Thompson introduced classes of orders generalizing the class of scattered orders.
For a regular cardinal \(\kappa\), they defined the classes of \(\kappa\)-scattered and
weakly \(\kappa\)-scattered linear orders. For \(\kappa = \omega\), these two classes
coincide and are equal to the classical class of scattered orders. For larger values
of \(\kappa\), though, the two classes are provably different. In this talk, we will
investigate properties of these generalized scattered orders with respect to
partition relations, in particular the extent to which the classes of
\(\kappa\)-scattered or weakly \(\kappa\)-scattered linear orders of size \(\kappa\) are
closed under partition relations of the form \(\tau \rightarrow (\varphi, n)\) for all natural
numbers \(n\). Along the way, we will prove a generalization of the Milner-Rado
paradox and look at some results regarding ordinal partition relations.
This is joint work with Thilo Weinert.
Trees with ascent paths. (Hebrew University of Jerusalem Logic Seminar, March 2017.)
The notion of an ascent path through a tree,
isolated by Laver, is a generalization of the notion of a cofinal branch and,
in many cases, the existence of an ascent path through a tree provides a concrete
obstruction to the tree being special. We will discuss some recent results
regarding ascent paths through \(\kappa\)-trees, where \(\kappa > \omega_1\) is a regular
cardinal. We will discuss the consistency of the existence or non-existence of a
special \(\mu^+\)-tree with a \(\mathrm{cf}(\mu)\)-ascent path, where \(\mu\) is a singular cardinal.
We will also discuss the consistency of the statement, "There are \(\omega_2\)-Aronszajn trees
but every \(\omega_2\)-tree contains an \(\omega\)-ascent path." We will connect these
topics with various square principles and with results about the productivity of chain conditions.
Pseudo-Prikry sequences.
(Arctic Set Theory 3. Kilpisjärvi, Finland, January 2017.)
(Slides) Reflections on the coloring and chromatic numbers. (Hebrew University of Jerusalem Logic Seminar, January 2017.)
Compactness phenomena play a central role in modern set theory, and
the investigation of compactness and incompactness for the coloring and chromatic
numbers of graphs has been a thriving area of research since the mid-20th century,
when De Bruijn and Erdős published their compactness theorem for finite chromatic
numbers. In this talk, we will briefly review some of the highlights in this area
and then present new results indicating, firstly, that the coloring number can only
exhibit a limit amount of incompactness, and, secondly, that large amounts of incompactness
for the chromatic number are compatible with strong compactness statements, including compactness
for the coloring number. This is joint work with Assaf Rinot.
Squares, stationary reflection, and incompactness. (Young Researcher's Seminar Week. Centre de Recerca Matemàtica, Barcelona, November 2016.)
The study of the tension between compactness and incompactness has played a central role in modern set theory. In this talk, we will consider the interplay between various compactness and incompactness principles, focusing in particular on square principles, stationary reflection, and Aronszajn trees. We will begin by discussing joint work with Yair Hayut regarding the extent to which the existence of weak square sequences is compatible with simultaneous stationary reflection, establishing a tight connection between these two notions. We then discuss applications of this work to higher Souslin trees and incompactness for the chromatic number of graphs.
Square sequences and simultaneous stationary reflection.
(SETTOP 2016. Fruska Gora, Serbia, June 2016.)
(Slides)
(Joint work with Yair Hayut)
There has been much work done in set theory investigating the
tension between compactness phenomena, such as stationary reflection,
and incompactness phenomena, such as Jensen's square principle.
It is a folklore result that, for a cardinal \(\mu\), \(\square_\mu\) implies a strong failure of
stationary reflection at \(\mu^+\), while, for a regular cardinal \(\kappa\), Todorcevic's
square principle \(\square(\kappa)\) implies the failure of simultaneous stationary
reflection for pairs of stationary subsets of \(\kappa\). We obtain
results about the extent to which weakenings of \(\square(\kappa)\) are
compatible with simultaneous stationary reflection at \(\kappa\).
Robust reflection principles. (ASL Winter Meeting, Seattle, January 2016.)
(Slides)
It is a common motif in set theory that, if \(\kappa\)
is a large cardinal (Mahlo, weakly compact, measurable, supercompact, etc.),
then \(\kappa\) satisfies certain interesting reflection principles. In addition,
because most large cardinal notions are preserved under small forcing extensions,
i.e. forcing extensions by posets with cardinality less than \(\kappa\), these
reflection principles, when they hold at large cardinals, are also robust under
small forcing. It has been a fruitful line of research to consider the extent
to which these reflection principles can hold at smaller cardinals. However,
these principles can in general fail to be preserved by small forcing when they
hold at small cardinals. Focusing on stationary reflection and the tree property,
we discuss situations in which reflection principles can fail to be robust under
small forcing, introduce natural strengthenings which are implied by large
cardinals and which are in all cases robust under small forcing, and consider the
extent to which these strenthenings can hold at small cardinals.
Robust reflection principles. (Cornell University Logic Seminar, September 2015.)
Large cardinals are useful in set theory in part
because they imply certain reflection properties. For example, if \(\kappa\) is a
weakly compact cardinal, then \(\kappa\) satisfies the tree property and every stationary
subset of \(\kappa\) reflects. An interesting direction of research involves
investigating the extent to which these reflection properties can hold at smaller
cardinals. By results of Levy and Solovay, for most large cardinal notions
(inaccessible, weakly compact, measurable, etc.), if \(\kappa\) is such a large cardinal
in V and \(\mathbb{P}\) is a forcing poset of size less than \(\kappa\), then \(\kappa\) remains
large after forcing with \(\mathbb{P}\). Therefore, reflection properties of a cardinal
\(\kappa\) that are implied by \(\kappa\) being a particular large cardinal are themselves
indestructible by small forcing when \(\kappa\) is such a large cardinal. However,
these reflection properties may no longer be indestructible at \(\kappa\) if \(\kappa\)
is a small cardinal. For example, it is consistent that \(\aleph_{\omega+1}\) has the
tree property but there is a forcing \(\mathbb{P}\) of size \(\omega_1\) such that \(\aleph_{\omega+1}\)
fails to have the tree property after forcing with \(\mathbb{P}\). We will begin by reviewing
the relevant large cardinal and reflection notions and will then consider
strengthenings of certain reflection properties that are always indestructible
under small forcing, focusing in particular on stationary reflection and the
tree property. We will look at the extent to which these reflection properties
can hold at small cardinals and the extent to which they are in fact stronger
than the weaker principles from which they are derived.
Coloring classes and the Hanf number for amalgamation.
(Ben Gurion University Logic, Set Theory, and Topology Seminar, March 2015.)
The amalgamation property is a topic of fundamental
importance in model theory and is still imperfectly understood. In the 1980s,
Grossberg asked a question, which remains open, about the existence of a Hanf
number for amalgamation in abstract elementary classes. In particular, Grossberg
conjectured that the Hanf number for amalgamation for classes given by an
\(\mathrm{L}_{\omega_1, \omega}\) sentence is \(\beth_{\omega_1}\). We introduce a new collection of
abstract elementary classes, called coloring classes, and use them to give a
partial answer to Grossberg's question, significantly improving upon work of
Baldwin, Kolesnikov, and Shelah. Analysis of these coloring classes leads to
some purely combinatorial questions that are of interest in their own right.
This is joint work with Alexei Kolesnikov.
Patterns of stationary reflection. (Winter School in Abstract Analysis,
Set Theory and Topology Section. Hejnice, Czech Republic. February 2015.)
(Slides)
If \(\kappa\) is a regular cardinal, \(\alpha < \kappa\) has
uncountable cofinality, and \(S\) is a stationary subset of \(\kappa\), we say that \(S\) reflects
at \(\alpha\) if \(S \cap \alpha\) is stationary in \(\alpha\). \(S\) reflects if there is
\(\alpha < \kappa\) such that \(S\) reflects at \(\alpha\). Questions regarding the extent of
stationary reflection have been extensively studied and are intimately related
to a number of topics concerning large cardinals, combinatorial set theory, and
cardinal arithmetic. Eisworth, motivated in part by his work on square-bracket
partition relations, asked whether it must be the case that if \(\lambda\) is a singular
cardinal and every stationary subset of \(\lambda^+\) reflects, then every stationary
subset of \(\lambda^+\) reflects at ordinals of arbitrarily high cofinality below
\(\lambda\). We will answer this in the negative and go on to consider variants of
Eisworth's question. Along the way, we will explore some connections between
stationary reflection and the combinatorial notion of approachability.
Coloring classes and amalgamation.
(Ariel University Seminar on Algorithms, Combinatorics, Graph Theory and Algebra, December 2014.)
The amalgamation property for a class of models is the
assertion that, under certain conditions, two models in the class can be realized
as sub-models of a single larger model in the class. The amalgamation property
for first-order logic is a useful consequence of the compactness theorem, but
it may fail for generalizations of first-order logic. In the 1980s, Grossberg
made a conjecture regarding the extent of amalgamation for models of sentences
in infinitary logics. We introduce a class of combinatorial structures called
well-colorings, prove results about the existence of well-colorings of certain
cardinalities, and use these results to show the optimality of Grossberg's
conjecture, improving upon results of Baldwin, Kolesnikov, and Shelah. This is
joint work with Alexei Kolesnikov.
Bounded stationary reflection.
(Hebrew University of Jerusalem Students' Set Theory Seminar, December 2014.)
Eisworth, motivated in part by his work on square-bracket
partition relations, asked whether it must be the case that if \(\mu\) is a singular
cardinal and every stationary subset of \(\mu^+\) reflects, then every stationary subset
of \(\mu^+\) reflects at an ordinal of arbitrarily high cofinality below \(\mu\). We will
give some background motivation, sketch a proof answering this question in the
negative, and discuss various variations on Eisworth's question.
Well-colorings and the Hanf number for amalgamation. (Bar-Ilan University Infinite Combinatorics Seminar, November 2014.)
The amalgamation property is a topic of fundamental interest in model theory and is still imperfectly understood. In the 1980s, Grossberg asked a question, which remains open to this day, about the existence of a Hanf number for amalgamation in abstract elementary classes. We introduce a new class of structures, called well-colorings, and use them to give a partial answer to Grossberg's question, significantly improving upon previous work of Baldwin, Kolesnikov, and Shelah. We shall start the talk by briefly discussing the relevant model-theoretic definitions and will then give proofs of the main results, which are entirely set-theoretic and combinatorial in nature and of interest in their own right. This is joint work with Alexei Kolesnikov.
Bounded stationary reflection. (Graduate Student Conference in Logic, April 2014.)
(Slides)
We will discuss the effects of square-bracket partition
relations on stationary reflection at the successors of singular cardinals. We
will then sketch a proof of the result that, relative to large cardinals, it is
consistent that there is a singular cardinal \(\mu\) such that every stationary subset of
\(\mu^+\) reflects but there is a stationary subset of \(\mu^+\) that does not reflect at
ordinals of arbitrarily high cofinality below \(\mu\). This answers a question of Todd
Eisworth and is joint work with James Cummings.
Jónsson cardinals, partition relations, and stationary reflection, part I. (CMU Logic Seminar, February 2014.)
I will give an introduction to Jónsson cardinals and related square bracket
partition relations. We will prove some of the basic facts about Jónsson cardinals,
focusing in particular on the important open question of whether the successor of a
singular cardinal can be Jónsson. This will involve a discussion of the connections
between Jónsson cardinals and stationary reflection, which will lead into a recent result of Cummings and myself.
Jónsson cardinals, partition relations, and stationary reflection, parts II and III.
(CMU Logic Seminar, February/March 2014.)
I will present a proof that, relative to large cardinal assumptions,
it is consistent that there is a singular cardinal
\(\mu\) such that every stationary subset of \(\mu^+\) reflects but there is a stationary
subset of \(\mu^+\) that does not reflect at
ordinals of arbitrarily high cofinality. This answers a question of Eisworth motivated
by the study of Jónsson cardinals and
square-bracket partition relations and is joint work with James Cummings.
The transfinite subway and closure properties of uncountable cardinals. (CMU Graduate Student Seminar, January 2014.)
It is a little-known fact that there is a subway line with uncountably many stops connecting the Hilbert Hotel to its nearest airport. In this talk, we will analyze the behavior and efficiency of this subway line. In the process, we will develop some of the combinatorial theory of uncountable cardinals and, time permitting, construct every Borel subset of the real numbers.