Characterizing large cardinals in terms of layered posets

Given an uncountable regular cardinal $\kappa$, a partial order is $\kappa$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $\kappa$ is stationary in $\mathcal{P}_\kappa(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $\kappa$ is weakly compact if and only if every partial order satisfying the $\kappa$-chain condition is $\kappa$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $\kappa$-Knaster partial orders are $\kappa$-stationarily layered implies that $\kappa$ is a Mahlo cardinal and every stationary subset of $\kappa$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal

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 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 $\square(\kappa)$

Layered posets and Kunen's universal collapse

We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $\kappa$ is weakly compact then any universal Kunen iteration of $\kappa$-cc posets (each possibly of size $\kappa$) is $\kappa$-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications: (1) For any $n \ge 1$, a wide variety of $<\omega_{n-1}$-closed, $\omega_{n+1}$-cc posets of size $\omega_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $\omega_n$ (see Theorem 7 and Corollary 8); and (2) For any $n \in \omega$, the Tree Property at $\omega_{n+3}$ is consistent with the Chang's Conjecture $(\omega_{n+3}, \omega_{n+1}) \twoheadrightarrow (\omega_{n+1}, \omega_n)$ (Theorem 9).Comment: corrected proof of Lemma

Knaster and friends I: Closed colorings and precalibers

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

Characterizations of the weakly compact ideal on PκλP_\kappa\lambda

Hellsten \cite{MR2026390} gave a characterization of $\Pi^1_n$-indescribable subsets of a $\Pi^1_n$-indescribable cardinal in terms of a natural filter base: when $\kappa$ is a $\Pi^1_n$-indescribable cardinal, a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq \kappa$. We generalize Hellsten's characterization to $\Pi^1_n$-indescribable subsets of $P_\kappa\lambda$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $\Pi^1_0$-indescribability ideal on $P_\kappa\lambda$ equals the minimal \emph{strongly} normal ideal $\text{NSS}_{\kappa,\lambda}$ on $P_\kappa\lambda$, and is not equal to $\text{NS}_{\kappa,\lambda}$ as may be expected, we formulate a notion of $n$-club subset of $P_\kappa\lambda$ and prove that a set $S\subseteq P_\kappa\lambda$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq P_\kappa\lambda$. We also prove that elementary embeddings considered by Schanker \cite{MR2989393} witnessing \emph{near supercompactness} lead to the definition of a normal ideal on $P_\kappa\lambda$, and indeed, this ideal is equal to Baumgartner's ideal of non--$\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$. Additionally, as applications of these results we answer a question of Cox-L\"ucke \cite{MR3620068} about $\mathcal{F}$-layered posets, provide a characterization of $\Pi^m_n$-indescribable subsets of $P_\kappa\lambda$ in terms of generic elementary embeddings, prove several results involving a two-cardinal weakly compact diamond principle and observe that a result of Pereira \cite{MR3640048} yeilds the consistency of the existence of a $(\kappa,\kappa^+)$-semimorasses $\mu\subseteq P_\kappa\kappa^+$ which is $\Pi^1_n$-indescribable for all $n<\omega$.Comment: revised version for APA

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 $\theta<\kappa$, 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

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