9 found
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  1.  41
    Continuous Ramsey theory on polish spaces and covering the plane by functions.Stefan Geschke, Martin Goldstern & Menachem Kojman - 2004 - Journal of Mathematical Logic 4 (2):109-145.
    We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max, which satisfy [Formula: see text] and prove: Theorem. For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see (...)
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  2.  23
    A dual open coloring axiom.Stefan Geschke - 2006 - Annals of Pure and Applied Logic 140 (1):40-51.
    We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of 1-dense real order types, Ann. Pure Appl. Logic 29 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in (...)
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  3.  50
    On the weak Freese–Nation property of ?(ω).Sakaé Fuchino, Stefan Geschke & Lajos Soukupe - 2001 - Archive for Mathematical Logic 40 (6):425-435.
    Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many (...)
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  4.  26
    On the weak Freese-Nation property of complete Boolean algebras.Sakaé Fuchino, Stefan Geschke, Saharon Shelah & Lajos Soukup - 2001 - Annals of Pure and Applied Logic 110 (1-3):89-105.
    The following results are proved: In a model obtained by adding ℵ 2 Cohen reals , there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. Modulo the consistency strength of a supercompact cardinal , the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. If a weak form of □ μ and cof =μ + hold for each μ >cf= ω , then the weak Freese-Nation property of 〈 P (...)
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  5.  77
    Applications of elementary submodels in general topology.Stefan Geschke - 2002 - Synthese 133 (1-2):31 - 41.
    Elementary submodels of some initial segment of the set-theoretic universe are useful in order to prove certain theorems in general topology as well as in algebra. As an illustration we give proofs of two theorems due to Arkhangelskii concerning cardinal invariants of compact spaces.
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  6.  21
    Low-distortion embeddings of infinite metric spaces into the real line.Stefan Geschke - 2009 - Annals of Pure and Applied Logic 157 (2-3):148-160.
    We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K>1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K>1, K-bi-Lipschitz embed into the real line.
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  7.  29
    Potential continuity of colorings.Stefan Geschke - 2008 - Archive for Mathematical Logic 47 (6):567-578.
    We say that a coloring ${c: [\kappa]^n\to 2}$ is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some ${\aleph_1}$ -preserving extension of the set-theoretic universe. Given an arbitrary coloring ${c:[\kappa]^n\to 2}$ , we define a forcing notion ${\mathbb P_c}$ that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that ${\mathbb P_c}$ is c.c.c. if and only if c (...)
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  8.  10
    The number of openly generated Boolean algebras.Stefan Geschke & Saharon Shelah - 2008 - Journal of Symbolic Logic 73 (1):151-164.
    This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly ϭ-filtered Boolean algebras. We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > N1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free. Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of (...)
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  9.  20
    Weak Borel chromatic numbers.Stefan Geschke - 2011 - Mathematical Logic Quarterly 57 (1):5-13.
    Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a (...)
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