Related papers: Full Souslin trees at small cardinals

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…

For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined,…

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$…

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and…

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $\kappa$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both…

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…

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed…

It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to…

By an omega_1 --tree we mean a tree of size omega_1 and height omega_1. An omega_1 --tree is called a Kurepa tree if all its levels are countable and it has more than omega_1 branches. An omega_1 --tree is called a Jech--Kunen tree if it…

In this note, we obtain an upper bound on the maximum number of distinct non-empty palindromes in starlike trees. This bound implies, in particular, that there are at most $4n$ distinct non-empty palindromes in a starlike tree with three…

It is well-known that the consistency strength of the GCH failing at a measurable cardinal is the existence of a cardinal $\kappa$ with $o(\kappa)=\kappa^{++}$. As the literature does not contain more than a proof sketch of the lower bound…

We determine the large cardinal consistency strength of the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$. Indeed, we show that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that…

In this paper we analyse some questions concerning trees on $\kappa$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in…

In this paper we analyze the connection between some properties of partially strongly compact cardinals: the completion of filters of certain size and instances of the compactness of $\mathcal{L}_{\kappa,\kappa}$. Using this equivalence we…

Let $\kappa$ be any regular cardinal. Assuming the existence of a huge cardinal above $\kappa$, we prove the consistency of $\binom{\kappa^{++}}{\kappa^+}\rightarrow\binom{\tau}{\kappa^+}$ for every ordinal $\tau<\kappa^{++}$. Likewise, we…

The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable…

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $\kappa = \kappa^{<\kappa} \geq\aleph_1$…

We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.

A space X is kappa-resolvable (resp. almost kappa-resolvable) if it contains kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X). Answering a problem raised by Juhasz, Soukup, and…

This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $\phi$, in the sense that $\phi$ characterizes $\kappa$, if…