Related papers: On $\Sigma_1^1$-completeness of quasi-orders on $\…

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular the isomorphism relation of dense…

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $\kappa$. We show the consistency of…

A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which…

We prove that for every uncountable cardinal $\kappa$ such that $\kappa^{<\kappa}=\kappa$, the quasi-order of embeddability on the $\kappa$-space of $\kappa$-sized graphs Borel reduces to the embeddability on the $\kappa$-space of…

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the…

We consider the following dichotomy for $\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa$: either all $R$-independent sets are of size at most $\kappa$, or there is a $\kappa$-perfect…

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…

We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on…

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $ of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies c.c.c.)…

We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $\hbox{NS} \upharpoonright A$ is $\aleph_2$-saturated, for $A$ a stationary co-stationary subset of…

We study the Borel-reducibility of isomorphism relations in the generalized Baire space $\kappa^\kappa$. In the main result we show for inaccessible $\kappa$, that if $T$ is a classifiable theory and $T'$ is superstable with the strong…

Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa…

We study $\Sigma_1(\omega_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $\Sigma_1$-formula with parameter $\omega_1$) in the presence of large cardinals. Our results show that the existence…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

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 find a natural $L_{\omega_1,\omega}$-axiomatisation $\Sigma$ of a structure on the upper half-plane $\mathbb{H}$ as the covering space of modular curves. The main theorem states that $\Sigma$ has a unique model in every uncountable…

We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is…

Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the…

We present an overview of results on the question of whether the non-stationary ideal of an uncountable regular cardinal $\kappa$ can be defined by a $\Pi_1$-formula using parameters of hereditary cardinality at most $\kappa$. These results…

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…