David Milovich

Friedman et al. (2013) considered the question whether model existence of L-omega 1,(omega)-sentences is absolute for transitive models of ZFC, in the sense that if V subset of W are transitive models of ZFC with the same ordinals, phi is an element of V and V satisfies"phi is an L-omega 1,(omega)-sentence", then V satisfies Phi if and only if W satisfies Phi where Phi is a first-order sentence with parameters phi and alpha asserting that phi has a model of size aleph(alpha). From Friedman et al. (2013) we know that the answer is positive for alpha = 0, 1, and under the negation of CH the answer is negative for all alpha > 1. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each alpha > 1, except the case when alpha =omega which is an open question in Friedman et al. (2013). We answer the open question by providing a negative answer under GCH even for alpha= omega. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all alpha > 1 assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in Friedman et al. (2013) to a Mahlo cardinal. Finally, we consider the absoluteness question for the aleph(alpha)-amalgamation property of L-omega 1,(omega)-sentences (under substructure). We prove that assuming GCH, aleph(alpha)-amalgamation is non-absolute for 1 < alpha < omega. This answers a question of Sinapova and Souldatos (2017). The cases alpha = 1 and a infinite remain open. As a corollary we show that it is non-absolute that the amalgamation spectrum of an L-omega 1,(omega)-sentence is empty.

For a continuous self-map of a star graph to be Li-Yorke chaotic and to have full periodicity, we prove some new sufficient conditions on the orbit of the center.

We show that there is a boolean algebra that has the Freese-Nation property (FN) but not the strong Freese-Nation property (SFN), thus answering a question of Heindorf and Shapiro. Along the way, we produce some new characterizations of the FN and SFN in terms of sequences of elementary submodels.

We establish negative results about "rectangular" local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type \omega x \omega_2. For another, the compactum \beta\omega has no nontrivially rectangular local bases, and the same is consistently true of \beta\omega \ \omega: no local base in \beta\omega has cofinal type \kappa x c if \kappa < m_{\sigma-n-linked} for some n in [1,\omega). Also, CH implies that every local base in \beta\omega \ \omega has the same cofinal type as one in \beta\omega. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on \omega always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on \omega is commutative modulo cofinal equivalence.

We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite \kappa, the (\kappa,\kappa)-FN, the (\kappa^+,\kappa)-FN, and the \kappa-FN are known to be equivalent; we show that the family of properties (\lambda,\mu)-FN for \lambda>\mu form a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (\kappa,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in terms of special subsets of boolean algebras. We show that the (\kappa,\aleph_0)-FN also leads to a generalization of the equality of weight and \pi-character in dyadic compacta. Elementary subalgebras and their duals, elementary quotient spaces, were originally used to define the (\lambda, \kappa)-FN and its topological dual, which naturally generalized from Stone spaces to all compacta, thereby generalizing Shchepin's notion of openly generated compacta. We introduce a simple combinatorial definition of the (\lambda, \kappa)-FN that is equivalent to the original for regular infinite cardinals \lambda>\kappa.