Combinatorial aspects of measure and category

Bartoszyński, Tomek

doi:10.4064/fm-127-3-225-239

Cited by 78 publications

(49 citation statements)

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“…As we have shown in the previous two lemmas, (1) implies both (5) and (6). We will prove that (5) implies (4), and (6) implies (2).…”

Section: Lemma 1 Let κ Be a Cardinal The Following Conditions Are Ementioning

confidence: 54%

Covering for category and trigonometric thin sets

Eliáš

2003

Proc. Amer. Math. Soc.

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Abstract. In this work we consider several combinatorial principles satisfied for cardinals smaller than cov(M), the covering number of the ideal of first category sets on real line. Using these principles we prove that there exist N 0 -sets (similarly N-sets, A-sets) which cannot be covered by fewer than cov(M) pD-sets (A-sets, N-sets, respectively). This improves the results of our previous paper (1997).

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“…As we have shown in the previous two lemmas, (1) implies both (5) and (6). We will prove that (5) implies (4), and (6) implies (2).…”

Section: Lemma 1 Let κ Be a Cardinal The Following Conditions Are Ementioning

confidence: 54%

Covering for category and trigonometric thin sets

Eliáš

2003

Proc. Amer. Math. Soc.

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“…Also, mc may be characterized as the maximum cardinal X such that, given any family of functions H of size < X , we may find a function g such that V/z e H, 3n e co g(n) = h(n). Some of the basic properties of this cardinal, including proofs of the equivalence of these characterizations, may be found in [1], [15], and [22]. It is clear from the above discussion that mc < d < c.…”

Section: On the Generic Existence Of Special Ultrafilters R Michael mentioning

confidence: 99%

On the generic existence of special ultrafilters

Canjar¹

1990

Proc. Amer. Math. Soc.

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Abstract.We introduce the concept of the generic existence of P-point, Qpoint, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically iff semiselectives do iff mc = c, and we show that ß-point ultrafilters exist generically iff semi-ß-points do iff mc -d , where d is the minimal cardinality of a dominating family of functions and m is the minimal cardinality of a cover of the real line by nowhere-dense sets. These results complement a result of Ketonen, that P-points exist generically iff c = d , and one of P. Nyikos and D. H. Fremlin, that saturated ultrafilters exist generically iff mc = c = 2 g(n). Following [21] we let d denote the minimum cardinality of a dominating family of functions. We let c be the cardinality a'o) and mc be the minimum cardinality of a cover of the real line consisting of nowhere-dense sets. The notation mc comes from "Martin's Axiom for countable partial orders"; mc is the maximum cardinal X such that the following holds: given any countable partial order P and fewer than X dense subsets of P, there exists a generic G C P which meets each of the dense subsets.

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“…Traces for functions from ω to ω were first introduced in set theory by Bartoszyński (see [2]), where he called them slaloms. He used them for forcing results related to cardinal characteristics of the continuum.…”

Section: Introductionmentioning

confidence: 99%

Lowness for Demuth Randomness

Downey

2009

Mathematical Theory and Computational Practice

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Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

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Combinatorial aspects of measure and category

Cited by 78 publications

References 0 publications

Covering for category and trigonometric thin sets

Covering for category and trigonometric thin sets

On the generic existence of special ultrafilters

Lowness for Demuth Randomness

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