arXiv subjects

Chris Lambie-Hanson

Publications and source records attributed to Chris Lambie-Hanson.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneously nonvanishing higher derived limits

The derived functors $\lim^n$ of the inverse limit find many applications in algebra and topology. In particular, the vanishing of certain derived limits $\lim^n \mathbf{A}[H]$, parametrized by an abelian group $H$, has implications for strong homology and condensed mathematics. In this paper, we prove that if $\mathfrak{d}=ω_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(ω_n)}$ (i.e. the direct sum of $ω_n$-many copies of $\mathbb{Z}$). The same holds for $H=\mathbb{Z}$ under the assumption that $\mathrm{w}\diamondsuit(S^{k+1}_k)$ holds for all $k < n$. In particular, this shows that if $\lim^n \mathbf{A}[H] = 0$ holds for all $n \geq 1$ and all abelian groups $H$, then $2^{\aleph_0} \geq \aleph_{ω+1}$, thus answering a question of Bannister. Finally, we prove some consistency results regarding simultaneous nonvanishing of derived limits, again in the case of $H = \mathbb{Z}$. In particular, we show the consistency, relative to $\mathsf{ZFC}$, of $\bigwedge_{2 \leq k < ω} \lim^k \mathbf{A} \neq 0$.

math.LO↗

arXiv2026

Generalized almost disjoint families and injective Banach spaces

A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an \emph{almost disjoint family} on a topological space $X$, generalizing the classical notion of almost disjoint families of subsets of $\mathbb{N}$, which we feel is of interest in its own right. We prove that, if $\mathfrak{b} = 2^{\aleph_0}$, then there exists an almost disjoint family of cardinality $2^{\aleph_1}$ on the Čech-Stone remainder of $\mathbb{N}$.

math.FA↗

arXiv2026

Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at $ω_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size $ω_1$. We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at $ω_2$.

math.LO↗

arXiv2026

Strongly increasing sequences

Using a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $ω_{2}$ consisting of functions from $ω$ to $ω$. We also show that there can be no such sequence of length $ω_{4}$.

math.LO↗

arXiv2026

Higher walks and squares

We continue the development of the theory of higher dimensional walks on ordinals began recently by Bergfalk. In particular we identify natural coherence conditions on higher dimensional $C$-sequences that entail coherence of the resultant higher rho-functions. We also introduce various higher square principles by adding non-triviality conditions to these coherent higher $C$-sequences and investigate basic properties of said square principles. For example, in analogy with the classical case, we prove that these higher square principles abound in the constructible universe but can be forced to fail, modulo large cardinals. Finally, we prove that certain higher rho-functions obtained by walking along higher square sequences exhibit non-triviality in addition to coherence. In particular, it follows that higher square principles on a cardinal $λ$ entail certain non-vanishing Čech cohomology groups for $λ$ considered with the order topology.

math.LO↗

arXiv2026

Preservation of some topological properties under forcing

We add to the theory of preservation of topological properties under forcing. In particular, we answer a question of Gilton and Holshouser in a strong sense, showing that if player II has a winning strategy in the strong countable fan tightness game of a space at a point, then this continues to hold in every set forcing extension of the universe. The same is also true for the Rothberger game, but not for the countable fan tightness or Menger games.

math.LO↗

arXiv2025

Whitehead's problem and condensed mathematics

One of the better-known independence results in general mathematics is Shelah's solution to Whitehead's problem of whether $\mathrm{Ext}^1(A,\mathbb{Z})=0$ implies that an abelian group $A$ is free. The point of departure for the present work is Clausen and Scholze's proof that, in contrast, one natural interpretation of Whitehead's problem within their recently-developed framework of condensed mathematics has an affirmative answer in $\mathsf{ZFC}$. We record two alternative proofs of this result, as well as several original variations on it, both for their intrinsic interest and as a springboard for a broader study of the relations between condensed mathematics and set theoretic forcing. We show more particularly how the condensation $\underline{X}$ of any locally compact Hausdorff space $X$ may be viewed as an organized presentation of the forcing names for the points of canonical interpretations of $X$ in all possible set-forcing extensions of the universe, and we argue our main result by way of this fact. We show also that when interpreted within the category of light condensed abelian groups, Whitehead's problem is again independent of the $\mathsf{ZFC}$ axioms. In fact we show that it is consistent that Whitehead's problem has a negative solution within the category of $κ$-condensed abelian groups for every uncountable cardinal $κ$, but that this scenario, in turn, is inconsistent with the existence of a strongly compact cardinal.

math.LO↗

arXiv2025

Infinitary combinatorics in condensed math and strong homology

Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse, pertaining, for example, to the sheaf theory of extremally disconnected spaces, to Banach--Smith duality, to the productivity of compact projective condensed anima, and to the structure of the derived category of condensed abelian groups. Underlying each of these implications are the combinatorics of multidimensionally coherent families of functions of small infinite cardinal height, and it is for this reason that we convene accounts of them together herein.

math.AT↗

arXiv2024

Kurepa trees, continuous images, and perfect set properties

Building upon work of Lücke and Schlicht, we study (higher) Kurepa trees through the lens of higher descriptive set theory, focusing in particular on various perfect set properties and representations of sets of branches through trees as continuous images of function spaces. Answering a question of Lücke and Schlicht, we prove that it is consistent with $\mathsf{CH}$ that there exist $ω_2$-Kurepa trees and yet, for every $ω_2$-Kurepa tree $T \subseteq {^{<ω_2}}ω_2$, the set $[T] \subseteq {^{ω_2}}ω_2$ of cofinal branches through $T$ is not a continuous image of ${^{ω_2}}ω_2$. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of ${^{ω_1}}ω_1$ failing to satisfy strong perfect set properties, and prove a number of consistency results regarding \emph{full} and \emph{superthin} trees.

math.LO↗

arXiv2024

Two-cardinal derived topologies, indescribability and Ramseyness

We introduce a natural two-cardinal version of Bagaria's sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Peter Holy and Philip White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

math.LO↗

arXiv2024

Squares, ultrafilters and forcing axioms

We study relationships between various set theoretic compactness principles, focusing on the interplay between the three families of combinatorial objects or principles mentioned in the title. Specifically, we show the following. (1) Strong forcing axioms, in general incompatible with the existence of indexed squares, can be made compatible with weaker versions of indexed squares. (2) Indexed squares and indecomposable ultrafilters with suitable parameters can coexist. As a consequence, the amount of stationary reflection known to be implied by the existence of a uniform indecomposable ultrafilter is optimal. (3) The Proper Forcing Axiom implies that any cardinal carrying a uniform indecomposable ultrafilter is either measurable or a supremum of countably many measurable cardinals. Leveraging insights from the preceding sections, we demonstrate that the conclusion cannot be improved.

math.LO↗

arXiv2023

Strong tree properties, Kurepa trees, and guessing models

We investigate the generalized tree properties and guessing model properties introduced by Weiß and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other prominent combinatorial principles. We introduce a weakening of Viale and Weiß's Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almsot Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular $θ\geq ω_2$, there are stationarily many $ω_2$-guessing models $M \in \mathscr{P}_{ω_2} H(θ)$ that are not $ω_1$-guessing models.

math.LO↗

arXiv2023

Hajnal--Máté graphs, Cohen reals, and disjoint type guessing

A Hajnal--Máté graph is an uncountably chromatic graph on $ω_1$ satisfying a certain natural sparseness condition. We investigate Hajnal-Máté graphs and generalizations thereof, focusing on the existence of Hajnal-Máté graphs in models resulting from adding a single Cohen real. In particular, answering a question of Dániel Soukup, we show that such models necessarily contain triangle-free Hajnal-Máté graphs. In the process, we isolate a weakening of club guessing called \emph{disjoint type guessing} that we feel is of interest in its own right. We show that disjoint type guessing is independent of $\mathsf{ZFC}$ and, if disjoint type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal-Máté graphs $G$ such that the chromatic numbers of finite subgraphs of $G$ grow arbitrarily slowly.

math.LO↗

arXiv2023

Narrow systems revisited

Motivated by two open questions about two-cardinal tree properties, we introduce and study generalized narrow system properties. The first of these questions asks whether the strong tree property at a regular cardinal $κ\geq ω_2$ implies the Singular Cardinals Hypothesis ($\mathsf{SCH}$) above $κ$. We show here that a certain narrow system property at $κ$ that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply $\mathsf{SCH}$ above $κ$. The second of these questions asks whether the strong tree property can consistenty hold simultaneously at all regular cardinals $κ\geq ω_2$. We show here that the analogous question about the generalized narrow system property has a positive answer. We also highlight some connections between generalized narrow system properties and the existence of certain strongly unbounded subadditive colorings.

math.LO↗

arXiv2023

Guessing models, trees, and cardinal arithmetic

Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is as small as possible relative to the value of $2^ω$. Building on work of Laver, we prove that, in the extension of any model of $\mathsf{PFA}$ by a measure algebra, every tree of height and size $ω_1$ is B-special (a generalization of specialness introduced by Baumgartner that can also hold of trees with uncountable branches). Finally, we investigate the impact of forcing axioms for Suslin and almost Suslin trees on guessing model properties. In particular, we prove thatif $S$ is a Suslin tree, then the axioms $\mathsf{PFA}(S)$ and $\mathsf{PFA}(S)[S]$ imply the Guessing Model Property and the Indestructible Guessing Model Property, respectively, and, if $T^*$ is an almost Suslin Aronszajn tree, then the axiom $\mathsf{PFA}(T^*)$ implies the Indestructible Guessing Model Property. This answers a number of questions of Cox and Krueger.

math.LO↗

arXiv2023

A Galvin-Hajnal theorem for generalized cardinal characteristics

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.

math.LO↗

arXiv2022

Polish space partition principles and the Halpern-Läuchli theorem

The Halpern-Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern-Läuchli theorem, and the same forcing from Harrington's proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

math.LO↗

arXiv2022

A note on highly connected and well-connected Ramsey theory

We study a pair of weakenings of the classical partition relation $ν\rightarrow (μ)^2_λ$ recently introduced by Bergfalk-Hrušák-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $ν$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.

math.LO↗