Authors:Arthur W. Apter Pages: 715 - 723 Abstract: We show that the theories “ZFC \(+\) There is a supercompact cardinal” and “ZFC \(+\) There is a supercompact cardinal \(+\) Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0538-7 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Fred Galvin; Jan Mycielski; Robert M. Solovay Pages: 725 - 732 Abstract: We show that strong measure zero sets (in a \(\sigma \) -totally bounded metric space) can be characterized by the nonexistence of a winning strategy in a certain infinite game. We use this characterization to give a proof of the well known fact, originally conjectured by K. Prikry, that every dense \(G_\delta \) subset of the real line contains a translate of every strong measure zero set. We also derive a related result which answers a question of J. Fickett. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0541-z Issue No:Vol. 56, No. 7-8 (2017)

Authors:Natasha Dobrinen; José G. Mijares; Timothy Trujillo Pages: 733 - 782 Abstract: A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not \(k+1\) -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \(\mathcal {P}(n)\) . If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \([\omega ]^{<\omega }\) . In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0540-0 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Assaf Rinot Pages: 783 - 796 Abstract: May the same graph admit two different chromatic numbers in two different universes' How about infinitely many different values' and can this be achieved without changing the cardinals structure' In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal \(\mu \) below the first fixed-point of the \(\aleph \) -function, there exists a graph \(\mathcal G_\mu \) satisfying the following: \(\mathcal G_\mu \) has size and chromatic number \(\mu \) ; for every infinite cardinal \(\kappa <\mu \) , there exists a cofinality-preserving \({{\mathrm{GCH}}}\) -preserving forcing extension in which \({{\mathrm{Chr}}}(\mathcal G_\mu )=\kappa \) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0551-x Issue No:Vol. 56, No. 7-8 (2017)

Authors:David Asperó Pages: 797 - 810 Abstract: I define a homogeneous \(\aleph _2\) –c.c. proper product forcing for adding many clubs of \(\omega _1\) with finite conditions. I use this forcing to build models of \(\mathfrak {b}(\omega _1)=\aleph _2\) , together with \(\mathfrak {d}(\omega _1)\) and \(2^{\aleph _0}\) large and with very strong failures of club guessing at \(\omega _1\) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0539-6 Issue No:Vol. 56, No. 7-8 (2017)

Authors:C. Laflamme; M. Pouzet; R. Woodrow Pages: 811 - 829 Abstract: Two structures are said to be equimorphic if each embeds in the other. Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case of linear orders, here also called chains. In particular we provide structure results for chains having less than continuum many isomorphism classes of equimorphic chains. We deduce as a corollary that any chain has either a single isomorphism class of equimorphic chains or infinitely many. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0545-8 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Michael Hrušák Pages: 831 - 847 Abstract: We study the Katětov order on Borel ideals. We prove two structural theorems (dichotomies), one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0543-x Issue No:Vol. 56, No. 7-8 (2017)

Authors:Piotr Koszmider Pages: 849 - 876 Abstract: We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper (Baumgartner and Shelah in Ann Pure Appl Logic 33(2):109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \(f:[\lambda ^{++}]^2\rightarrow [\lambda ^{++}]^{\le \lambda }\) with the appropriate \(\lambda ^+\) -version of property \(\Delta \) for regular \(\lambda \ge \omega \) satisfying \(\lambda ^{<\lambda }=\lambda \) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0544-9 Issue No:Vol. 56, No. 7-8 (2017)

Authors:J. A. Larson Pages: 877 - 909 Abstract: James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0546-7 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Pierre Matet; Saharon Shelah Pages: 911 - 934 Abstract: We give a new characterization of the nonstationary ideal on \(P_\kappa (\lambda )\) in the case when \(\kappa \) is a regular uncountable cardinal and \(\lambda \) a singular strong limit cardinal of cofinality at least \(\kappa \) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0552-9 Issue No:Vol. 56, No. 7-8 (2017)

Authors:William J. Mitchell Pages: 935 - 982 Abstract: Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \(\kappa \) having an extender of length \(\kappa ^{+\omega _1}\) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0547-6 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Itay Neeman Pages: 983 - 1036 Abstract: We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \(\omega _2\) . The first involves adding a \(\Box _{\omega _1}\) sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \(\omega _2\) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0550-y Issue No:Vol. 56, No. 7-8 (2017)

Authors:Tadatoshi Miyamoto Pages: 1037 - 1044 Abstract: We show that a coding principle introduced by J. Moore with respect to all ladder systems is equiconsistent with the existence of a strongly inaccessible cardinal. We also show that a coding principle introduced by S. Todorcevic has consistency strength at least of a strongly inaccessible cardinal. PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0548-5 Issue No:Vol. 56, No. 7-8 (2017)

Authors:Justin Tatch Moore; Stevo Todorcevic Pages: 1105 - 1114 Abstract: In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \(2^{\aleph _0} \ge \aleph _2\) and every pair of \(\aleph _2\) -dense subsets of \(\mathbb {R}\) are isomorphic as linear orders. The main result is the isolation of a combinatorial principle \((**)\) which is immune to c.c.c. forcing and which in the presence of \(2^{\aleph _0} \le \aleph _2\) implies that two \(\aleph _2\) -dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC that there exists an \(\aleph _2\) dense suborder X of \(\mathbb {R}\) which cannot be embedded into \(-X\) in any outer model with the same \(\aleph _2\) . PubDate: 2017-11-01 DOI: 10.1007/s00153-017-0549-4 Issue No:Vol. 56, No. 7-8 (2017)

Authors:H. Pinedo; C. Uzcategui Abstract: We show that the enveloping space \({\mathbb {X}}_G\) of a partial action of a Polish group G on a Polish space \({\mathbb {X}}\) is a standard Borel space, that is to say, there is a topology \(\tau \) on \({\mathbb {X}}_G\) such that \(({\mathbb {X}}_G, \tau )\) is Polish and the quotient Borel structure on \({\mathbb {X}}_G\) is equal to \(Borel({\mathbb {X}}_G,\tau )\) . To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught’s transform are valid for partial actions of groups. PubDate: 2017-11-02 DOI: 10.1007/s00153-017-0598-8

Authors:Jan Mycielski; Grzegorz Tomkowicz Abstract: We show that several theorems about Polish spaces, which depend on the axiom of choice ( \(\mathcal {AC}\) ), have interesting corollaries that are theorems of the theory \(\mathcal {ZF} + \mathcal {DC}\) , where \(\mathcal {DC}\) is the axiom of dependent choices. Surprisingly it is natural to use the full \(\mathcal {AC}\) to prove the existence of these proofs; in fact we do not even know the proofs in \(\mathcal {ZF} + \mathcal {DC}\) . Let \(\mathcal {AD}\) denote the axiom of determinacy. We show also, in the theory \(\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})\) , a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in \(\mathcal {ZFC}\) , but assuming the existence of large enough cardinals it is true in the model \(\langle L(\mathbb {R}),\in \rangle \) . PubDate: 2017-10-29 DOI: 10.1007/s00153-017-0596-x

Authors:Joel David Hamkins; Thomas A. Johnstone Abstract: We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing. PubDate: 2017-08-19 DOI: 10.1007/s00153-017-0542-y