Related papers: Mutually Generics and Perfect Free Subsets

We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact…

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbf{\Delta}^1_2$ maximal independent set. As a main…

Let $G_0$,..., $G_{n-1}$ be mutually generic over $V$, each $G_i$ adding at least one new real over $V$. We show that the transcendence degree of the reals of $V[G_0, \dots, G_{n-1}]$ is maximal (of size continuum) over the field generated…

For graphs F and G an F-matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F. The number of F-matchings in G is denoted by s(F,G). We show that for every fixed positive integer m and every fixed tree F, the…

Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at…

We study the relationship between Amoeba forcing (the partial order which generically adds a measure one set of random reals) and projective measurability. Given a universe V of set theory and a forcing notion P in V we say that V is…

I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot…

We prove a result on perfect cliques with respect to countably many G-delta relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any…

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing.…

In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC example of a Hausdorff space in $C(\omega_1)$ that has…

We investigate families of subsets of $\omega$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $\omega$. More precisely, we study the following topics and questions: 1) Examples of…

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less…

Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb{R}^{d})$ called a potential we define its rotation set $R(F)$ as the set of integrals of $F$ with respect to all $T$-invariant…

It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set $X$ of finitely many Cohen reals, by showing that, in the forcing…

Let $\ee>0$ and $\fff$ be a family of finite subsets of the Cantor set $\ccc$. Following D. H. Fremlin, we say that $\fff$ is $\ee$-filling over $\ccc$ if $\fff$ is hereditary and for every $F\subseteq\ccc$ finite there exists $G\subseteq…

We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice…

It is well known from universal algebra that, for every signature $\Sigma$, there exist algebras over $\Sigma$ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping…

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector, where $\delta_{\mathcal{F}}(I)$ ($\delta_{\mathcal{N}}(I)$, respectively) is the facet…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…