Transformations of the transfinite plane

We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .

Freely indecomposable almost free groups with free abelianization

For certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L. This strengthens an earlier result of Eklof and Mekler.

UNCOUNTABLE TREES AND COHEN -REALS

We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

UNIVERSAL CLASSES NEAR ${\aleph _1}$

Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

SOME APPLICATIONS OF SUPERCOMPACT EXTENDER BASED FORCINGS TO HOD

Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model where all regular uncountable cardinals are measurable in HOD is constructed.