Related papers: Partition properties for simply definable colourin…

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{\omega_2}]^2\to 2$ establishing $\omega_2 \not\to [(\omega_1;{\omega})]^2_2$ such that some colouring…

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2)…

The primary goal of this paper is to establish a model of $ZFC$ wherein the definable tree property is affirmed for all uncountable regular cardinals. This endeavor commences with the utilization of both a supercompact cardinal and a…

We get a quite maximal version of the colouring property $Pr_1$ by proving $Pr_1(\lambda,\lambda,\lambda,\theta)$ when $\lambda = \partial^+, \partial > \theta$ are regular cardinals.

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…

Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes,…

We study the possible structures which can be carried by sets which have no countable subset, but which fail to be `surjectively Dedekind finite', in two possible senses, that there is a surjection to $\omega$, or alternatively, that there…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

We characterize the compactness properties of the product of \lambda\ copies of the space \omega\ with the discrete topology, dealing in particular with the case \lambda\ singular, using regular and uniform ultrafilters, infinitary…

We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects…

A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of…

Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…

Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the…

Recent results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman's Theorem fail for all uncountable cardinals. Results in the positive direction were obtained by Komj\'ath, the…

A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol…

Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is…

We introduce various colouring principles which generalise the so-called "onto mapping principle" of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow to…

A set partition $\sigma$ of $[n]=\{1,\cdots ,n\}$ contains another set partition $\omega$ if a standardized restriction of $\sigma$ to a subset $S\subseteq[n]$ is equivalent to $\omega$. Otherwise, $\sigma$ avoids $\omega$. Sagan and Goyt…

We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.

We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal,…