Related papers: The tree property at successors of singular cardin…

We consider mainly the following version of set theory:"ZF + DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a…

We investigate the unbalanced ordinary partition relations of the form $\lambda \rightarrow {(\lambda, \alpha)}^{2}$ for various values of the cardinal $\lambda$ and the ordinal $\alpha$. For example, we show that for every infinite…

We show that the tree property, stationary reflection and the failure of approachability at $\kappa^{++}$ are consistent with $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$, where $\kappa$ is a singular strong limit cardinal with the…

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be…

Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong…

1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no…

We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit…

We show that it is consistent, relative to $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$.

We construct a large family of normal $\kappa$-complete $\mathbb{R}_\kappa$-embeddable non-special $\kappa^+$-Aronszajn trees which have no club isomorphic subtrees using an instance of the proxy principle of Brodsky-Rinot.

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…

We show that if $\kappa < \aleph_\omega$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ in the extension. Under some further hypotheses on the ground model,…

We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(\kappa^+,\kappa)\notcc(\aleph\_1,\aleph\_0)$ when $\kappa$ is supercompact. The…

Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komj\'ath and…

In this short note we show that if lambda>aleph_1 is regular and lambda is not the successor of a singular cardinal of cofinality aleph_0, and G is a lambda-free abelian group of size lambda, then there is a free group G' subseteq G of size…

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this…

The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M…

We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism…

Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not…

We answer a question of Usuba by showing that the combinatorial principle $UB_\lambda$ can fail at a singular cardinal. Furthermore, $\lambda$ can be taken to be $\aleph_\omega.$

Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$