Related papers: Selective Independence and $h$-Perfect Tree Forcin…

A family $\mathcal{A} \subseteq [\omega]^\omega$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is said to be…

We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field.…

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion…

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add…

We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…

In a self-contained way, we deal with revised countable support iterated forcing for the reals. We improve theorems on preservation of the property UP, weaker than semi proper, and we hopefully improve the presentation. We continue [Sh:b,…

We investigate properties of trees of height $\omega_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $\omega_1$-tree. We introduce fragments of subcompleteness which are…

Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving $\aleph_1$. In…

I introduce a new family of axioms extending ZFC set theory, the $\Sigma_n$-correct forcing axioms. These assert roughly that whenever a forcing name $\dot{a}$ can be forced by a poset in some forcing class $\Gamma$ to have some $\Sigma_n$…

For $f,g\in\omega\ho$ let $\mycfa_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e. for every branch $\nu$ of the $f$-tree, one of the $g$-trees contains $\nu$. $\myc_{f,g}$ is…

The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $\sigma$-continuity of a certain Borel not $\sigma$-continuous function. We give a characterization of this forcing in the language of trees and using this…

In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak…

Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erd\H{o}s-Ko-Rado Theorem to the…

The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property…

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma$: the hierarchy of bounded forcing axioms, of $\Sigma^1_1$-absoluteness and of Aronszajn tree preservation principles. The latter…

We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from…

It is well-known that extending the Hilbert axiomatic system for first-order intuitionistic logic with an exclusion operator, that is dual to implication, collapses the domains of models into a constant domain. This makes it an interesting…

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting…

We prove uniform consistency of Random Survival Forests (RSF), a newly introduced forest ensemble learner for analysis of right-censored survival data. Consistency is proven under general splitting rules, bootstrapping, and random selection…