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Hossein Lamei Ramandi

Publications and source records attributed to Hossein Lamei Ramandi.

8 recordsLinked to original sources
arXiv2023

Aronszajn Free Kurepa Trees

We consider a transitive relation on the power set of $ω_1$ 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 $ω_1$, then there are Kurepa trees which are not club isomorphic. These maximal subsets of $ω_1$ exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever $α_0 \in ω_1$ and $X \subset ω_1$ are such that $ω_1^{\textsc{L}[X \cap α_0]} = ω_1, ω_2^{\textsc{L}[X]} = ω_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.

math.LO↗
arXiv2023

A minimal Kurepa line

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 consistent with $\ZFC$ that there is a minimal Kurepa line which does not contain any Aronszajn suborder.

math.LO↗
arXiv2023

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 an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U \subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $\textrm{MA}_{ω_2}$ holds and $ω_2$ 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 $ρ$ function which might be useful in other contexts.

math.LO↗
arXiv2023

On the rigidity of Souslin trees and their generic branches

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 $C \subset ω_1$, $S\upharpoonright C$ is rigid. This answers Fuchs's questions in Club degrees of rigidity and almost Kurepa trees. Moreover, we show it is consistent with $\diamondsuit$ that for every Souslin tree there is a dense $X \subset S$ which does not have a copy of $S$. This is related to a question due to Baumgartner.

math.LO↗
arXiv2020

Galvin's Question on non-$σ$-Well Ordered Linear Orders

Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $ω_1$. We show it is consistent that $\mathcal{C}$ has minimal elements. This answers an old question due to Galvin.

math.LO↗
arXiv2017

There may be no minimal non $σ$-scattered linear orders

In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $σ$-scattered. This shows that a theorem of Laver, which asserts that the class of $σ$-scattered linear orders is well quasi-ordered, is sharp. We also prove that PFA${}^+$ implies that every non $σ$-scattered linear order either contains a real type, an Aronszajn type, or a ladder system indexed by a stationary subset of $ω_1$, equipped with either the lexicographic or reverse lexicographic order. Our work immediately implies that CH is consistent with "no Aronszajn tree has a base of cardinality $\aleph_1$." This gives an affirmative answer to a problem due to Baumgartner.

math.LO↗