Related papers: An abstract elementary class non-axiomatizable in …

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH…

In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{\kappa, \aleph_{0}}$ is categorical in a cardinal $\lambda > \kappa$, and $\kappa$ is a measurable cardinal. There we prove that the class of model of…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\lambda^+$-minimal types and…

Consider an a.e.c. (abstract elementary class), that is, a class K of models with a partial order refining inclusion (submodel) which satisfy the most basic properties of an elementary class. Our test question is trying to show that the…

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in…

We prove that for infinite cardinals $\kappa<\lambda$ the alternating group $Alt(\lambda)$ (of even permutations) of $\lambda$ is not embeddable into the symmetric group $Sym(\kappa)$ (of all permutations) of $\kappa$. To prove this fact we…

We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.

We prove that every abstract elementary class (a.e.c.) with LST number $\kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(\kappa)^{+++},\kappa^+}(\tau)$. In this logic an a.e.c.…

Let $\kappa$ be a regular cardinal, $\lambda<\kappa$ be a smaller infinite cardinal, and $\mathsf K$ be a $\kappa$-accessible category where colimits of $\lambda$-indexed chains exist. We show that various category-theoretic constructions…

For a cardinal kappa and a model M of cardinality kappa let No(M) denote the number of non-isomorphic models of cardinality kappa which are L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when kappa is a weakly compact…

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…

In this paper, we prove that if $\kappa$ is a almost strongly compact cardinal, then any MAEC with L\"owenheim-Skolem number below $\kappa$ is $<\kappa$-d-tame.

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $\lambda$ such that whenever $M, N \in K$ have…

Let \lambda be a cardinal with \lambda=\lambda^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality \lambda and characteristic p such that the automorphism group of K_0 is a free group of…

Suppose lambda is a singular cardinal of uncountable cofinality kappa. For a model M of cardinality lambda, let No(M) denote the number of isomorphism types of models N of cardinality lambda which are L_{infty lambda}-equivalent to M. In…

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length…

Let B(kappa, lambda) be the subalgebra of P(kappa) generated by [kappa]^{<= lambda}. It is shown that if B is any homomorphic image of B(kappa, lambda) then either |B|< 2^lambda or |B|=|B|^lambda, moreover if X is the Stone space of B then…

We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(\lambda_i : i \le…

For cardinals lambda, kappa, theta we consider the class of graphs of cardinality lambda which has no subgraph which is (kappa, theta)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak)…