4 research outputs found
A Rigid Kurepa Tree From a Free Suslin Tree
We analyze a countable support product of a free Suslin tree which turns it
into a highly rigid Kurepa tree with no Aronszajn subtree
On constructions with -cardinals
We propose developing the theory of consequences of morasses relevant in
mathematical applications in the language alternative to the usual one,
replacing commonly used structures by families of sets originating with
Velleman's neat simplified morasses called -cardinals. The theory of related
trees, gaps, colorings of pairs and forcing notions is reformulated and
sketched from a unifying point of view with the focus on the applicability to
constructions of mathematical structures like Boolean algebras, Banach spaces
or compact spaces.
A new result which we obtain as a side product is the consistency of the
existence of a function
with the
appropriate -version of property for regular
satisfying .Comment: Minor correction
Can You Take Komjath's Inaccessible Away?
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 is a Kurepa tree and every Kurepa tree contains a Souslin subtree,
if there is an inaccessible cardinal. This is stronger than Komjath's theorem
that asserts the same consistency from two inaccessible cardinals. We will show
that our large cardinal assumption is optimal, i.e. if every Kurepa tree has an
Aronszajn subtree then is inaccessible in the constructible universe
\textsc{L}. Moreover, we prove it is consistent with ZFC that there is a Kurepa
tree such that if is a Kurepa tree with the inherited order
from , then has an Aronszajn subtree. This theorem uses no large
cardinal assumption. Our last theorem immediately implies the following: assume
holds and is not a Mahlo cardinal in
\textsc{L}. Then there is a Kurepa tree with the property that every Kurepa
subset has an Aronszajn subtree. Our work entails proving a new lemma about
Todorcevic's function which might be useful in other contexts.Comment: 20 page