1,339 research outputs found
Robust reflection principles
A cardinal satisfies a property P robustly if, whenever
is a forcing poset and ,
satisfies P in . We study the extent to which certain
reflection properties of large cardinals can be satisfied robustly by small
cardinals. We focus in particular on stationary reflection and the tree
property, both of which can consistently hold but fail to be robust at small
cardinals. We introduce natural strengthenings of these principles which are
always robust and which hold at sufficiently large cardinals, consider the
extent to which these strengthenings are in fact stronger than the original
principles, and investigate the possibility of these strengthenings holding at
small cardinals, particularly at successors of singular cardinals.Comment: 20 page
On the growth rate of chromatic numbers of finite subgraphs
We prove that, for every function ,
there is a graph with uncountable chromatic number such that, for every with , every subgraph of with fewer than
vertices has chromatic number less than . This answers a question of
Erd\H{o}s, Hajnal, and Szemeredi.Comment: 10 page
Bounded stationary reflection II
Bounded stationary reflection at a cardinal is the assertion that
every stationary subset of reflects but there is a stationary subset
of that does not reflect at arbitrarily high cofinalities. We produce
a variety of models in which bounded stationary reflection holds. These include
models in which bounded stationary reflection holds at the successor of every
singular cardinal and models in which bounded stationary
reflection holds at but the approachability property fails at
Good and bad points in scales
We address three questions raised by Cummings and Foreman regarding a model
of Gitik and Sharon. We first analyze the PCF-theoretic structure of the
Gitik-Sharon model, determining the extent of good and bad scales. We then
classify the bad points of the bad scales existing in both the Gitik-Sharon
model and other models containing bad scales. Finally, we investigate the ideal
of subsets of singular cardinals of countable cofinality carrying good scales.Comment: 26 page
Pseudo-Prikry sequences
We generalize results of Gitik, Dzamonja-Shelah, and Magidor-Sinapova on the
existence of pseudo-Prikry sequences, which are sequences that approximate the
behavior of the generic objects introduced by Prikry-type forcings, in outer
models of set theory. Such sequences play an important role in the study of
singular cardinal combinatorics by placing restrictions on the type of behavior
that can consistently be obtained in outer models. In addition, we provide
results about the existence of diagonal pseudo-Prikry sequences, which
approximate the behavior of the generic objects introduced by diagonal
Prikry-type forcings. Our proof techniques are substantially different from
those of previous results and rely on an analysis of PCF-theoretic objects in
the outer model.Comment: 15 page
Squares and narrow systems
A narrow system is a combinatorial object introduced by Magidor and Shelah in
connection with work on the tree property at successors of singular cardinals.
In analogy to the tree property, a cardinal satisfies the \emph{narrow
system property} if every narrow system of height has a cofinal
branch. In this paper, we study connections between the narrow system property,
square principles, and forcing axioms. We prove, assuming large cardinals, both
that it is consistent that satisfies the narrow system
property and holds and that it
is consistent that every regular cardinal satisfies the narrow system property.
We introduce natural strengthenings of classical square principles and show how
they can be used to produce narrow systems with no cofinal branch. Finally, we
show that the Proper Forcing Axiom implies that every narrow system of
countable width has a cofinal branch but is consistent with the existence of a
narrow system of width with no cofinal branch.Comment: 26 page
Covering properties and square principles
Covering matrices were introduced by Viale in his proof that the Singular
Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of
his work and in subsequent work with Sharon, he isolated two reflection
principles, and , which may hold of covering
matrices. In this paper, we continue previous work of the author investigating
connections between failures of and and variations
on Jensen's square principle. We prove that, for a regular cardinal , assuming large cardinals, is consistent with
for all with . We
demonstrate how to force nice -covering matrices for which
fail to satisfy and . We investigate normal covering
matrices, showing that, for a regular uncountable ,
implies the existence of a normal -covering matrix for but
that cardinal arithmetic imposes limits on the existence of a normal
-covering matrix for when is uncountable. We also
investigate certain increasing sequences of functions which arise from covering
matrices and from PCF-theoretic considerations and show that a stationary
reflection hypothesis places limits on the behavior of these sequences.Comment: 22 page
Aronszajn trees, square principles, and stationary reflection
We investigate questions involving Aronszajn trees, square principles, and
stationary reflection. We first consider two strengthenings of
introduced by Brodsky and Rinot for the purpose of
constructing -Souslin trees. Answering a question of Rinot, we prove
that the weaker of these strengthenings is compatible with stationary
reflection at but the stronger is not. We then prove that, if is
a singular cardinal, implies the existence of a special
-tree with a -ascent path, thus answering a question
of L\"ucke
Knaster and friends I: Closed colorings and precalibers
The productivity of the -chain condition, where is a
regular, uncountable cardinal, has been the focus of a great deal of
set-theoretic research. In the 1970s, consistent examples of -cc posets
whose squares are not -cc were constructed by Laver, Galvin, Roitman
and Fleissner. Later, examples were constructed by Todorcevic,
Shelah, and others. The most difficult case, that in which ,
was resolved by Shelah in 1997.
In this work, we obtain analogous results regarding the infinite productivity
of strong chain conditions, such as the Knaster property. Among other results,
for any successor cardinal , we produce a example of a
poset with precaliber whose power is not
-cc. To do so, we carry out a systematic study of colorings satisfying
a strong unboundedness condition. We prove a number of results indicating
circumstances under which such colorings exist, in particular focusing on cases
in which these colorings are moreover closed
A forcing axiom deciding the generalized Souslin Hypothesis
We derive a forcing axiom from the conjunction of square and diamond, and
present a few applications, primary among them being the existence of
super-Souslin trees. It follows that for every uncountable cardinal ,
if is not a Mahlo cardinal in G\"odel's constructible universe,
then entails the existence of a -complete
-Souslin tree
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