arXiv subjects
Publications and source records attributed to Ari Meir Brodsky.
We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $κ$ and a positive integer $n$, a $κ$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.
The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper, as new foundations for the construction of $κ$-Souslin trees in a uniform way that does not depend on the nature of the (regular uncountable) cardinal $κ$. Since their introduction, these principles have facilitated construction of Souslin trees with complex combinations of features, and have enabled the discovery of completely new scenarios in which Souslin trees must exist. Furthermore, the proxy principles have found new applications beyond the construction of trees. This paper opens with a comprehensive exposition of the proxy principles. We motivate their very definition, emphasizing the utility of each of the parameters and the consequent flexibility that they provide. We then survey the findings surrounding them, presenting a rich spectrum of unrelated models and configurations in which the proxy principles are known to hold, and showcasing a gallery of Souslin trees constructed from the principles. The last two sections of the paper offer new results. In particular, for every positive integer $n$, we give a construction of a $λ^+$-Souslin tree all of whose $n$-derived trees are Souslin, but whose $(n+1)$-power is special.
In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of $\diamondsuit$. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved.
We work with a pre-$λ$-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $λ$. We investigate the density of uniqueness triples in a given pre-$λ$-frame $\mathfrak s$, that is, under what circumstances every basic triple admits a non-forking extension that is a uniqueness triple. Prior results in this direction required strong hypotheses on $\mathfrak s$. Our main result is an improvement, in that we assume far fewer hypotheses on $\mathfrak s$. In particular, we do not require $\mathfrak s$ to satisfy the extension, uniqueness, stability, or symmetry properties, or any form of local character, though we do impose the amalgamation and stability properties in $λ^+$, and we do assume $\diamondsuit(λ^+)$. As a corollary, by applying our main result to the trivial $λ$-frame, it follows that in any AEC $\mathbf K$ satisfying modest hypotheses on $\mathbf K_λ$ and $\mathbf K_{λ^+}$, the set of $*$-domination triples in $\mathbf K_λ$ is dense among the non-algebraic triples. We also apply our main result to the non-splitting relation, obtaining the density of uniqueness triples from very few hypotheses.
Ben-David and Shelah proved that if $λ$ is a singular strong-limit cardinal and $2^λ=λ^+$, then $\square^*_λ$ entails the existence of a normal $λ$-distributive $λ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_λ$ by $\square(λ^+,{<}λ)$. As $\square(λ^+,{<}λ)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $κ$ regular uncountable, $\square(κ)$ entails the existence of a partition of $κ$ into $κ$ many fat sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that $ω_2$ cannot be split into two fat sets.
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.
We propose a parameterized proxy principle from which $κ$-Souslin trees with various additional features can be constructed, regardless of the identity of $κ$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $κ$-Souslin tree that applies also for $κ$ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known $\diamondsuit$-based constructions of $κ$-Souslin trees may be redirected through this new proxy principle.
We study the relationship between a $κ$-Souslin tree $T$ and its reduced powers $T^θ/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $θ$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $ T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.
Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $κ$ be any infinite regular cardinal, let $ξ$ be any ordinal such that $2^{\left|ξ\right|} < κ$, and let $k$ be any natural number. Then \[ \text{non-$\left(2^{<κ}\right)$-special tree } \to \left(κ+ ξ\right)^2_k. \] This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<κ})^+$, the simplest example of a non-$(2^{<κ})$-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem: Let $κ$ be any infinite regular cardinal, let $ξ$ be any ordinal such that $2^{\left|ξ\right|} < κ$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<κ})^1_{2^{<κ}}$. Then \[ P \to \left(κ+ ξ\right)^2_k. \]