arXiv subjects

Chris Lambie-Hanson

Publications and source records attributed to Chris Lambie-Hanson.

46 records · Page 3Linked to original sources

arXiv2017

Squares, ascent paths, and chain conditions

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević's principle $\square(κ)$ implies an indexed version of $\square(κ,λ)$, we show that for all infinite, regular cardinals $λ<κ$, the principle $\square(κ)$ implies the existence of a $κ$-Aronszajn tree containing a $λ$-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which $\aleph_2$-Aronszajn trees exist and all such trees contain $\aleph_0$-ascent paths. Finally, we use our techniques to show that the assumption that the $κ$-Knaster property is countably productive and the assumption that every $κ$-Knaster partial order is $κ$-stationarily layered both imply the failure of $\square(κ)$.

math.LO↗

arXiv2017

Reflection on the coloring and chromatic numbers

We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado's conjecture, Fodor-type reflection, $Δ$-reflection, Stationary-sets reflection, Martin's Maximum, and a generalized Chang's conjecture. This is accomplished by showing that, under $\mathrm{GCH}$-type assumptions, instances of incompactness for the chromatic number can be derived from square-like principles that are compatible with large amounts of compactness. In addition, we prove that, in contrast to the chromatic number, the coloring number does not admit arbitrarily large incompactness gaps.

math.LO↗

arXiv2017

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 $κ$ satisfies the \emph{narrow system property} if every narrow system of height $κ$ 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 $\aleph_{ω+1}$ satisfies the narrow system property and $\square_{\aleph_ω, < \aleph_ω}$ 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 $ω_1$ with no cofinal branch.

math.LO↗

arXiv2016

Aronszajn trees, square principles, and stationary reflection

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(κ)$ introduced by Brodsky and Rinot for the purpose of constructing $κ$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $κ$ but the stronger is not. We then prove that, if $μ$ is a singular cardinal, $\square_μ$ implies the existence of a special $μ^+$-tree with a $\mathrm{cf}(μ)$-ascent path, thus answering a question of Lücke.

math.LO↗

arXiv2016

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, $\mathrm{CP}$ and $\mathrm{S}$, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of $\mathrm{CP}$ and $\mathrm{S}$ and variations on Jensen's square principle. We prove that, for a regular cardinal $λ> ω_1$, assuming large cardinals, $\square(λ, 2)$ is consistent with $\mathrm{CP}(λ, θ)$ for all $θ$ with $θ^+ < λ$. We demonstrate how to force nice $θ$-covering matrices for $λ$ which fail to satisfy $\mathrm{CP}$ and $\mathrm{S}$. We investigate normal covering matrices, showing that, for a regular uncountable $κ$, $\square_κ$ implies the existence of a normal $ω$-covering matrix for $κ^+$ but that cardinal arithmetic imposes limits on the existence of a normal $θ$-covering matrix for $κ^+$ when $θ$ 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.

math.LO↗

arXiv2015

Robust reflection principles

A cardinal $λ$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < λ$, $λ$ satisfies P in $V^{\mathbb{Q}}$. 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.

math.LO↗

arXiv2015

The Hanf number for amalgamation of coloring classes

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by showing, in ZFC, that the (disjoint) amalgamation property for classes $K_α$ studied in that paper must hold up to $\beth_α$ (only a consistency result was previously known).

math.LO↗

arXiv2015

Bounded stationary reflection II

Bounded stationary reflection at a cardinal $λ$ is the assertion that every stationary subset of $λ$ reflects but there is a stationary subset of $λ$ 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 $μ> \aleph_ω$ and models in which bounded stationary reflection holds at $μ^+$ but the approachability property fails at $μ$.

math.LO↗

arXiv2014

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.

math.LO↗

arXiv2013

Squares and Covering Matrices

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof amd subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between $\square_κ$ and $\square(κ^+)$ for a regular cardinal $κ$. We provide a detailed picture of the implications between these square principles.

math.LO↗