Related papers: Almost Souslin Kurepa trees
It is consistent that there exists a Souslin tree $T$ such that after forcing with it, $T$ becomes an almost Souslin Kurepa tree. This answers a question of Zakrzewski.
Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at $\omega_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing…
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there…
In this note we answer a question of Kunen (15.13 [Mi91]) showing that it is consistent that there are full Souslin trees
We prove that it is consistent that there exists a Kurepa tree $T$ such that ${}^{\omega_1}2$ is a continuous image of the topological space $[T]$ consisting of all cofinal branches of $T$ with respect to the cone topologies. This result…
We show it is consistent that there is a Souslin tree $S$ such that after forcing with $S$, $S$ is Kurepa and for all clubs $C \subset \omega_1$, $S\upharpoonright C$ is rigid. This answers Fuchs's questions in Club degrees of rigidity and…
We show it is consistent with $\ZFC$ that there is an everywhere Kurepa line which is order isomorphic to all of its dense $\aleph_2$-dense suborders. Moreover, this Kurepa line does not contain any Aronszajn suborder. We also show it is…
For uncountable downwards closed subtrees $U$ and $W$ of an $\omega_1$-tree $T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection is a finite union of countable chains. The tree $T$ is strongly non-saturated if…
We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of $ZFC$ $+$ $CH$. In this paper we present a sufficient condition (for sets of cardinals) to be consistently the set of cardinalities of branches…
Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the…
In this paper we investigate the problem of the distributivity of Kurepa trees. We show that it is consistent that there are Kurepa trees and for every Kurepa tree there is a small forcing notion which adds a branch to it without collapsing…
A $\kappa$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\kappa$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit…
A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be…
We introduce an abstract framework for forcing over a free Suslin tree with suborders of products of forcings which add some structure to the tree using countable approximations. The main ideas of this framework are consistency, separation,…
We analyze a countable support product of a free Suslin tree which turns it into a highly rigid Kurepa tree with no Aronszajn subtree. In the process, we introduce a new rigidity property for trees, which says roughly speaking that any…
We will show it is consistent with $GCH$ that there is a minimal Kurepa tree with respect to club embeddings.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
We introduce a variant of the Kurepa family. We then use one such family to construct a ccc indestructible property associated with a complete coherent Suslin tree $S$. Moreover, in every ccc forcing extension that preserves Suslin of $S$,…
Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary…
We consider a transitive relation on the power set of $\omega_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of…