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Chris Lambie-Hanson

Publications and source records attributed to Chris Lambie-Hanson.

At least 37 records · Page 2Linked to original sources
arXiv2022

Indestructibility of some compactness principles over models of PFA

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree property, $\mathsf{ISP}$, is preserved by adding any number of Cohen subsets of $ω$. And moreover, $\mathsf{GMP}$ implies that no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $κ= κ^{<κ}$ (we denote this principle $\mathsf{GMP}_{κ^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{ω+1}$-Kurepa trees and no $\aleph_{ω+2}$-Aronszajn trees.

math.LO↗
arXiv2022

Higher-dimensional Delta-systems

We investigate higher-dimensional $Δ$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $Δ$-system lemma. We focus in particular on systems that consist of sets of ordinals, in which case useful order-theoretic uniformities can be ensured. We then present three applications of these higher-dimensional $Δ$-systems to problems involving the interplay between forcing and partition relations on the reals.

math.LO↗
arXiv2021

Knaster and friends III: Subadditive colorings

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $θ<κ$, the existence of a strongly unbounded coloring $c:[κ]^2 \rightarrowθ$ is a theorem of ZFC. Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring is independent of ZFC. We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $κ$-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring is equivalent to a certain weak indexed square principle. We conclude the paper with an application to the failure of the infinite productivity of $κ$-stationarily layered posets, answering a question of Cox.

math.LO↗
arXiv2021

Simultaneously vanishing higher derived limits

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\mathbf{A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim^n\mathbf{A}$ (the $n^{\text{th}}$ derived limit of $\mathbf{A}$) vanishes for every $n >0$. Since that time, the question of whether it is consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for every $n >0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that, assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for all $n >0$. We show this via a finite support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration a condition equivalent to $\lim^n\mathbf{A}=0$ will hold for each $n>0$. This condition is of interest in its own right; namely, it is the triviality of every coherent $n$-dimensional family of certain specified sorts of partial functions $\mathbb{N}^2\to\mathbb{Z}$ which are indexed in turn by $n$-tuples of functions $f:\mathbb{N}\to\mathbb{N}$. The triviality and coherence in question here generalize the well-studied case of $n=1$.

math.LO↗
arXiv2021

Simutaneously vanishing higher derived limits without large cardinals

A question dating to Sibe Mardešić and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $\mathrm{lim}^n$ $(n>0)$ of a certain inverse system $\mathbf{A}$ indexed by ${^ω}ω$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^ω}ω$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $\beth_ω$-many Cohen reals, $\mathrm{lim}^n \mathbf{A}$ vanishes for all $n > 0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $Δ$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $\mathrm{lim}^n \mathbf{A}$ for all $n > 0$.

math.LO↗
arXiv2020

Diagonal supercompact Radin forcing

Motivated by the goal of constructing a model in which there are no $κ$-Aronszajn trees for any regular $κ>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.

math.LO↗
arXiv2020

Disjoint type graphs with no short odd cycles

In this note, we provide a proof of a technical result of Erdős and Hajnal about the existence of disjoint type graphs with no odd cycles. We also prove that this result is sharp in a certain sense.

math.LO↗
arXiv2020

Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erdős-Rado theorem for pairs is witnessed by the colouring $Δ_κ: [2^κ]^2 \rightarrow κ$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $Δ_κ$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $Δ$-regressive and almost $Δ$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $Δ_κ$ has the minimal cardinality of any \emph{maximal} triangle-free or odd-cycle-free colouring into $κ$. We resolve the question positively for odd-cycle-free colourings.

math.LO↗
arXiv2020

Separating diagonal stationary reflection principles

We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.

math.LO↗
arXiv2020

Forcing a $\square(κ)$-like principle to hold at a weakly compact cardinal

Hellsten \cite{MR2026390} proved that when $κ$ is $Π^1_n$-indescribable, the \emph{$n$-club} subsets of $κ$ provide a filter base for the $Π^1_n$-indescribability ideal, and hence can also be used to give a characterization of $Π^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteqκ$. By replacing clubs with $n$-clubs in the definition of $\Box(κ)$, one obtains a $\Box(κ)$-like principle $\Box_n(κ)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(κ)$ is consistent with the $Π^1_n$-indescribability of $κ$ but inconsistent with the $Π^1_{n+1}$-indescribability of $κ$. By generalizing the standard forcing to add a $\Box(κ)$-sequence, we show that if $κ$ is $κ^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $κ$ remains $κ^+$-weakly compact and $\Box_1(κ)$ holds. If $κ$ is $Π^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $κ$ is $κ^+$-weakly compact, $\Box_1(κ)$ holds and every weakly compact subset of $κ$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $Π^1_2$-indescribable cardinal, it is consistent that $κ$ is $κ^+$-weakly compact, every weakly compact subset of $κ$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $κ$ such that there is no $β<κ$ for which both $S^0\capβ$ and $S^1\capβ$ are weakly compact.

math.LO↗
arXiv2019

Knaster and friends II: The C-sequence number

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this series.

math.LO↗
arXiv2019

The Cohomology of the Ordinals I: Basic Theory and Consistency Results

In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. We show in particular that the Čech cohomology groups of the ordinals articulate higher-dimensional generalizations of Todorcevic's walks and coherent sequences techniques, and begin to account for those techniques' `unreasonable effectiveness' on $ω_1$. This discussion occupies the first half of our paper and is written with a general mathematical audience in mind. We turn in the paper's second half to more properly set-theoretic considerations. We describe a number of consistency results on the cohomology groups of the ordinals which certify their status as a graded family of incompactness principles. We show in particular that nontrivial cohomology groups on the ordinals are in some tension with large cardinals, and are maximally extant in Gödel's model $\mathrm{L}$. We describe forcings to add, then trivialize, nontrivial $n$-cocycles, and conclude with some comparison of these principles with those benchmark incompactness phenomena, the existence of square sequences and failures of stationary reflection.

math.LO↗
arXiv2019

On the growth rate of chromatic numbers of finite subgraphs

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices has chromatic number less than $k$. This answers a question of Erdős, Hajnal, and Szemeredi.

math.LO↗
arXiv2018

Knaster and friends I: Closed colorings and precalibers

The productivity of the $κ$-chain condition, where $κ$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $κ$-cc posets whose squares are not $κ$-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, $\mathsf{ZFC}$ examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $κ= \aleph_2$, 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 $κ$, we produce a $\mathsf{ZFC}$ example of a poset with precaliber $κ$ whose $ω^{\mathrm{th}}$ power is not $κ$-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.

math.LO↗
arXiv2018

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 $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.

math.LO↗
arXiv2018

Partitioning subsets of generalised scattered orders

In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erdős and Rado founded the partition calculus in a seminal paper. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. It provides analogues of the Milner-Rado paradox for these orders instead of ordinals. For infinite, regular $κ$, we investigate the extent to which the classes of $κ$-scattered, weakly $κ$-scattered, and $κ$-saturated linear orders of size $κ$ are closed under the partition relation $τ\rightarrow (φ, n)^2$ for all $n < ω$. We prove that for a regular cardinal $κ$ such that the stick principle holds at $κ$ and $\mathfrak{b}_κ= κ^+$, the partition relation $κ^+κ\rightarrow (κ^+κ, 3)^2$ fails. Finally we generalise a result of Komjáth and Shelah about partitions of scattered linear orders to a similar result about partitions of $κ$-scattered linear orders for uncountable $κ$. Together this continues older research by Erdős, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.

math.LO↗
arXiv2017

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.

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