Related papers: Partition properties for simply definable colourin…

In this note, we first discuss some properties of generated $\sigma$-fields and a simple approach to the construction of finite $\sigma$-fields. It is shown that the $\sigma$-field generated by a finite class of $\sigma$-distinct sets which…

We construct a measure on omega-one^2 over the ground model in the forcing extension of a measure algebra, and investigate when measure theoretic properties of some measurable colouring of omega-one^2 imply the existence of an uncountable…

We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions in \cite{MR2354904} and \cite{MR2902230}. In particular, we show for inaccessible $\kappa$,…

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$…

In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is…

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's…

A usual dichotomy is that in many cases, reasonably definable sets, satisfy the CH, i.e. if they are uncountable they have cardinality continuum. A strong dichotomy is when: if the cardinality is infinite it is continuum as in [Sh:273]. We…

We characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard…

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal…

Let $\kappa$ be any regular cardinal. Assuming the existence of a huge cardinal above $\kappa$, we prove the consistency of $\binom{\kappa^{++}}{\kappa^+}\rightarrow\binom{\tau}{\kappa^+}$ for every ordinal $\tau<\kappa^{++}$. Likewise, we…

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model…

Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence F and a parameter R subseteq H(kappa) such that functions…

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many…

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\delta$ cover with no continuum-sized ($G_\delta$)-dense subcollection. We also prove that in a countably compact weakly Lindel\"of normal…

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for…

We study the structured rainbow Ramsey theory at uncountable cardinals. When compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain…

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{\kappa, \kappa}$, this allows us to characterize any large cardinal…

We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_\theta,\mathfrak{p}_\theta,\mathfrak{g}_\theta,\mathfrak{r}_\theta,\mathfrak{t}_\theta$ at uncountable regular…

We will consider a number of new large-cardinal properties, the $\alpha$-tremendous cardinals for each limit ordinal $\alpha>0$, the hyper-tremendous cardinals, the $\alpha$-enormous cardinals for each limit ordinal $\alpha>0$, and the…