Related papers: Proper forcing and rectangular Ramsey theorems
We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…
The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a…
This is an exposition of much of Sections VI.3 and XVIII.3 of "Proper and Improper Forcing", including preservations for "no random reals over V", "reals of V form a non-meager set", "every dense open set contains a dense open set in V",…
We define the dualizations of objects and concepts which are essential for investigating the Ramsey property in the first levels of the projective hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and dual Laver…
We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the…
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are…
We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper aleph_2-c.c. forcing notions. We then look at some special cases (Q_D 's preceded by random forcing).
Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual…
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over…
This paper presents a type theory with a form of equality reflection: provable equalities can be used to coerce the type of a term. Coercions and other annotations, including implicit arguments, are dropped during reduction of terms. We…
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to…
We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply…
We examine a version of Ramsey's theorem based on Tao, Gaspar and Kohlenbach's "finitary" infinite pigeonhole principle.We will show that the "finitary" infinite Ramsey's theorem naturally gives rise to statements at the level of the…
We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl Theorem, the author's Ramsey lift of bowtie-free graphs as well…
A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic…
In this paper we provide explicit dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and…
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an…
We prove limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss…