More on Compactness of Chromatic Numbers

We prove that for any regular kappa and mu > kappa below the first fix point (lambda = aleph_lambda) above kappa, there is a graph with chromatic number > kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic number less than or equal to kappa.Comment: 8 page

Almost-free groups in varieties

AbstractLet V be a non-trivial variety of groups. For every 0 < n < ω, V has a non-free ℵn-free group of cardinality ℵn if and only V is not a variety of nilpotent groups of prime power exponent. If V is not a variety of nilpotent groups of prime power exponent then either V has a non-free κ-free group of power κ in every cardinal κ for which such an abelian group exists or V has a non-free κ+-free group of power κ+ for every cardinal κ

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Compactness of powers of \omega

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 elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.Comment: v2 slightly improve