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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., Comment: 47 pages; comments are welcome

Building upon work of L\"{u}cke 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 L\"{u}cke and Schlicht, we prove that it is consistent with $\mathsf{CH}$ that there exist $\omega_2$-Kurepa trees and yet, for every $\omega_2$-Kurepa tree $T \subseteq {^{<\omega_2}}\omega_2$, the set $[T] \subseteq {^{\omega_2}}\omega_2$ of cofinal branches through $T$ is not a continuous image of ${^{\omega_2}}\omega_2$. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of ${^{\omega_1}}\omega_1$ failing to satisfy strong perfect set properties, and prove a number of consistency results regarding \emph{full} and \emph{superthin} trees., Comment: 48 pages

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 $\mathfrak{d}=\omega_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(\omega_n)}$ (i.e. the direct sum of $\omega_n$-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 $2^{\aleph_0} \geq \aleph_{\omega+1}$, 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 $\bigwedge_{2 \leq k < \omega} \lim^k \mathbf{A} \neq 0$., Comment: 31 pages

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., Comment: 32 pages

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., Comment: Added citation to the work of Catalina Torres

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 $\kappa$-condensed abelian groups for every uncountable cardinal $\kappa$, but that this scenario, in turn, is inconsistent with the existence of a strongly compact cardinal., Comment: 40 pages, corrected statement of Theorem 6.3

A Hajnal--M\'{a}t\'{e} graph is an uncountably chromatic graph on $\omega_1$ satisfying a certain natural sparseness condition. We investigate Hajnal-M\'{a}t\'{e} graphs and generalizations thereof, focusing on the existence of Hajnal-M\'{a}t\'{e} graphs in models resulting from adding a single Cohen real. In particular, answering a question of D\'{a}niel Soukup, we show that such models necessarily contain triangle-free Hajnal-M\'{a}t\'{e} 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-M\'{a}t\'{e} graphs $G$ such that the chromatic numbers of finite subgraphs of $G$ grow arbitrarily slowly., Comment: 16 pages

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 $\kappa \geq \omega_2$ implies the Singular Cardinals Hypothesis ($\mathsf{SCH}$) above $\kappa$. We show here that a certain narrow system property at $\kappa$ that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply $\mathsf{SCH}$ above $\kappa$. The second of these questions asks whether the strong tree property can consistenty hold simultaneously at all regular cardinals $\kappa \geq \omega_2$. 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., Comment: 18 pages

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 $2^{\omega_1}$ is as small as possible relative to the value of $2^\omega$. 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 $\omega_1$ 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., Comment: 32 pages

The Halpern-L\"auchli 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-L\"auchli 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.

We investigate the generalized tree properties and guessing model properties introduced by Wei\ss\ 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 Wei\ss'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 $\theta \geq \omega_2$, there are stationarily many $\omega_2$-guessing models $M \in \mathscr{P}_{\omega_2} H(\theta)$ that are not $\omega_1$-guessing models., Comment: 35 pages

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $\omega$ will not add an $\omega_2$-Aronszajn tree or a weak $\omega_1$-Kurepa tree, and moreover no $\sigma$-centered forcing can add a weak $\omega_1$-Kurepa tree (a tree of height and size $\omega_1$ with at least $\omega_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $\omega_2$-Aronszajn or $\omega_1$-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 $\omega$. And moreover, $\mathsf{GMP}$ implies that no $\sigma$-centered forcing can add a weak $\omega_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $\kappa = \kappa^{<\kappa}$ (we denote this principle $\mathsf{GMP}_{\kappa^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{\omega+1}$-Kurepa trees and no $\aleph_{\omega+2}$-Aronszajn trees., Comment: 18 pages

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., Comment: 18 pages

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 $\theta<\kappa$, the existence of a strongly unbounded coloring $c:[\kappa]^2 \rightarrow\theta$ 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 $\kappa$-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 $\kappa$-stationarily layered posets, answering a question of Cox.

A question dating to Sibe Marde\v{s}i\'{c} 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 ${^\omega}\omega$ 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 ${^\omega}\omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $\beth_\omega$-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 $\Delta$-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 $n > 0$., Comment: 30 pages, 1 figure

We investigate higher-dimensional $\Delta$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $\Delta$-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 $\Delta$-systems to problems involving the interplay between forcing and partition relations on the reals., Comment: 24 pages

We study a pair of weakenings of the classical partition relation $\nu \rightarrow (\mu)^2_\lambda$ recently introduced by Bergfalk-Hru\v{s}\'{a}k-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $\nu$-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., Comment: 15 pages

In this note, we provide a proof of a technical result of Erd\H{o}s 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., Comment: 6 pages. Unpublished note

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., Comment: 29 pages

The optimality of the Erd\H{o}s-Rado theorem for pairs is witnessed by the colouring $\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$ 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 $\Delta_\kappa$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $\Delta$-regressive and almost $\Delta$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $\Delta_\kappa$ has the minimal cardinality of any \emph{maximal} triangle-free or odd-cycle-free colouring into $\kappa$. We resolve the question positively for odd-cycle-free colourings., Comment: 16 pages

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.

In 1988, Sibe Marde\v{s}i\'{c} 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 $n=1$., Comment: 35 pages. Accepted to Forum of Mathematics: Pi

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 Erd\H{o}s, Hajnal, and Szemeredi., Comment: 10 pages

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

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 \v{C}ech cohomology. We show in particular that the \v{C}ech 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 $\omega_1$. 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 G\"{o}del'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., Comment: 45 pages; revision of Lemma 2.22 (and several preparatory remarks) occasioned version 2

The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares are not $\kappa$-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 $\kappa = \aleph_2$, 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 $\kappa$, we produce a $\mathsf{ZFC}$ example of a poset with precaliber $\kappa$ whose $\omega^{\mathrm{th}}$ power is not $\kappa$-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.

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 Todor\v{c}evi\'{c}'s principle $\square(\kappa)$ implies an indexed version of $\square(\kappa,\lambda)$, we show that for all infinite, regular cardinals $\lambda<\kappa$, the principle $\square(\kappa)$ implies the existence of a $\kappa$-Aronszajn tree containing a $\lambda$-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 $\kappa$-Knaster property is countably productive and the assumption that every $\kappa$-Knaster partial order is $\kappa$-stationarily layered both imply the failure of $\square(\kappa)$.

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, $\Delta$-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.

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

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 pages

In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erd\H{o}s 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 $\kappa$, we investigate the extent to which the classes of $\kappa$-scattered, weakly $\kappa$-scattered, and $\kappa$-saturated linear orders of size $\kappa$ are closed under the partition relation $\tau \rightarrow (\varphi, n)^2$ for all $n < \omega$. We prove that for a regular cardinal $\kappa$ such that the stick principle holds at $\kappa$ and $\mathfrak{b}_\kappa = \kappa^+$, the partition relation $\kappa^+\kappa \rightarrow (\kappa^+\kappa, 3)^2$ fails. Finally we generalise a result of Komj\'{a}th and Shelah about partitions of scattered linear orders to a similar result about partitions of $\kappa$-scattered linear orders for uncountable $\kappa$. Together this continues older research by Erd\H{o}s, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, D\v{z}amonja, Komj\'{a}th, Shelah and Thompson.

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

Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail., Comment: 19 pages

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

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 $\lambda > \omega_1$, assuming large cardinals, $\square(\lambda, 2)$ is consistent with $\mathrm{CP}(\lambda, \theta)$ for all $\theta$ with $\theta^+ < \lambda$. We demonstrate how to force nice $\theta$-covering matrices for $\lambda$ which fail to satisfy $\mathrm{CP}$ and $\mathrm{S}$. We investigate normal covering matrices, showing that, for a regular uncountable $\kappa$, $\square_\kappa$ implies the existence of a normal $\omega$-covering matrix for $\kappa^+$ but that cardinal arithmetic imposes limits on the existence of a normal $\theta$-covering matrix for $\kappa^+$ when $\theta$ 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 pages

A cardinal $\lambda$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < \lambda$, $\lambda$ 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., Comment: 20 pages

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 $\kappa$ satisfies the \emph{narrow system property} if every narrow system of height $\kappa$ 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 $\aleph_{\omega+1}$ satisfies the narrow system property and $\square_{\aleph_{\omega}, < \aleph_{\omega}}$ 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 $\omega_1$ with no cofinal branch., Comment: 26 pages

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

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 $K_\alpha$ studied in that paper must hold up to $\beth_\alpha$ (only a consistency result was previously known)., Comment: 18 pages

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 pages

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 $\square_{\kappa}$ and $\square(\kappa^+)$ for a regular cardinal $\kappa$. We provide a detailed picture of the implications between these square principles., Comment: 24 pages