Related papers: Open Colorings and Baumgartner's Axiom

We contribute to the study of $\aleph_1$-dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is consistent with $\textsf{MA}$ that there exists an…

We consider several variants of Baumgartner's axiom for $\aleph_1$-dense sets defined on the Baire and Cantor spaces in terms of Lipschitz functions with respect to the usual metric. A variation of Baumgartner's original argument shows that…

We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{\aleph_0}=\aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah.…

It is proved that for each natural number $n$, if $\left| \mathbb{R} \right| = {\aleph}_{n}$, then there is a coloring of ${\left[ \mathbb{R} \right]}^{n+2}$ into ${\aleph}_{0}$ colors that takes all colors on ${\left[ X \right]}^{n+2}$…

This work is concerned with an axiom introduced by Todorc\v{e}vi\'{c} in \cite{stevo} that constitutes a Ramsey-like statement regarding the topology of the reals. Our aim is to explain the axiom in detail, give some interesting…

Addressing a question of Paul Larson we prove the following statement. If Chang's conjecture fails, Martin's axiom holds and the continuum is greater than $\aleph_2$, there are no weakly Laver ideals over $\aleph_1$. We also prove that…

We prove that if ZF is consistent then ZFC+GCH is consistent with the following statement: There is for every k<omega a model of cardinality aleph_1 which is L_{infty,omega_1}-equivalent to exactly k non-isomorphic models of cardinality…

We prove a better coloring theorem for aleph_4 and even aleph_3. This has a general topology consequence.

We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a…

We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $\kappa\geq\omega_2$ is a regular cardinal, we prove that there is a…

A subset of a model of ${\sf PA}$ is called neutral if it does not change the $\mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in…

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every…

Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $\kappa,\theta$, every colouring $c:G\longrightarrow\theta$ yields a subset $X\subseteq G$ with $|X|=\kappa$ such that $X+X$ is…

We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to…

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals and of the natural numbers. The theorems we are chiefly interested in assert the…

Abstractly, the generic extensions after $\aleph_\omega$-many Cohen reals and $\aleph_{\omega+1}$-many Cohen reals must be different for reasons of uniform density the relevant Boolean algebras. Nevertheless this is not satisfying and it…

Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for…

We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$…

We prove that if less than $\aleph_{\omega}$-many Cohen reals are added to a model of \textsf{CH}, then $\omega^{\ast}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, there is an ultrafilter on $\omega$ that does not…