Related papers: Square-bracket operations clubs
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…
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…
For Fr{\'e}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We…
We show that the set of codes for Ramsey positive analytic sets is $\mathbf{\Sigma}^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is…
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…
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the…
For every indecomposable ordinal $\alpha < \omega_1$, we introduce a variant of Abraham forcing for adding a club in $\omega_1$, which is $<\alpha$-proper but not $\alpha$-proper.
In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Ne\v{s}et\v{r}il and V. R\"{o}dl claims that the class of all finite posets…
Let $n\in\omega$. The weak choice principle $\operatorname{RC}_n$ states that for every infinite set $x$ there is an infinite subset $y\subseteq x$ with a choice function on $[y]^n:=\{z\subseteq y\mid \lvert z\rvert =n\}$.…
We discuss ways of adjoining perfect sets of mutually generic random reals. In particular, we show that if V \sub W are models of ZFC and W contains a dominating real over V, then W[r], where r is random over W, contains a perfect tree of…
We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp,…
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…
We show that several classical Ramseyan statements, and a forcing statement, are each equivalent to having Rothberger's property in all finite powers.
In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M.\ Soki\'c that the free product of Ramsey classes is a Ramsey class and the result of M.\ Bodirsky, M.\Pinsker and T.\…
The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish…
Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F)…
We continue the development of the theory of capturing schemes over $\omega_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum…
We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including $SL(n,\Z)$ for…
On this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups $G_\omega$ act without bounded orbit. Moreover, if the sequence $\omega$ does not contain…
We define a nontrivial version of the square principle $\Box_\omega$, which we then show consistent by means of forcing with finite conditions. This paper has been withdrawn by the author due to the fact that the presented $\Box_\omega$ can…