Related papers: Combinatorial principles from adding Cohen reals

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…

We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $\omega_2$ Cohen reals to a model of $\diamondsuit$.

Cohen's first model is a model of Zermelo--Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from…

We characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard…

Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. Old results are reviewed and a number of new results are…

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…

We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…

We give a generalization of Hochster's formula for local cohomologies of square-free monomial ideals to monomial ideals, which are not necessarily square-free. Using this formula, we give combinatorial characterizations of generalized…

Assuming the existence of a strong cardinal $\kappa$ and a measurable cardinal above it, we force a generic extension in which $\kappa$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds…

In the first part of the paper, we show that if $\omega \le \kappa < \lambda$ are cardinals, $\kappa^{<\kappa} = \kappa$, and $\lambda$ is weakly compact, then in $V[\M(\kappa,\lambda)]$ the tree property at $\lambda =…

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets…

We proved a truncated second main theorem of level one with explicit exceptional sets for analytic maps into $\mathbb P^2$ intersecting the coordinate lines with sufficiently high multiplicities. As applications, we studied some cases of…

We find new "reasons" for a class of models for not having a universal model in a cardinal $\lambda$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which…

A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented…

It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba (O. Kouba, A duality based proof of the Combinatorial Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a direct proof of the…

In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the…

We show that, assuming GCH, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and obeying the constraints of Easton's theorem, namely,…

We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $\kappa^{<\kappa}=\kappa$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to…

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $\omega$ will not add an $\omega_2$-Aronszajn tree or a weak $\omega_1$-Kurepa tree, and moreover no $\sigma$-centered forcing can add a…

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…