Filter games on ω and the dual ideal

Laflamme, Claude; Leary, Christopher C.

doi:10.4064/fm173-2-4

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) the Union A\ U A 2 Fails To Be Oand -Bounded For All Filters2

Proof Of Theorem 111

Ramsey and Convergence Properties1

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2006

2022

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Cited by 11 publications

(14 citation statements)

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“…The following claim is known, we give here a proof for reader's convenience. A stronger form of this result is presented in [17]. Claim 2.3.…”

Section: Proof Of Theorem 11mentioning

confidence: 86%

See 1 more Smart Citation

Mathias Forcing and Combinatorial Covering Properties of Filters

Chodounský¹,

Repovš²,

Zdomskyy³

2015

J. symb. log.

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“…The following claim is known, we give here a proof for reader's convenience. A stronger form of this result is presented in [17]. Claim 2.3.…”

Section: Proof Of Theorem 11mentioning

confidence: 86%

“…The formulation of[11, Problem 4] involves notions which will not be used in our paper, and hence we refer the reader to[11] for its precise formulation 4. In case X is a filter on ω, notice the similarity of this game to the game G(F + , [ω] <ω , F + ) from[17], where F = X <ω .…”

mentioning

confidence: 99%

Mathias Forcing and Combinatorial Covering Properties of Filters

Chodounský¹,

Repovš²,

Zdomskyy³

2015

J. symb. log.

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“…https://doi.org/10.1017/S1446788700014348 We shall reduce our game OFj? to the game (8(iT, M < w , 2f) considered in [17] and [18]. …”

Section: ) the Union A\ U A 2 Fails To Be Oand -Bounded For All Filtersmentioning

confidence: 99%

“…Now let us pass to proofs of our results. We start from the proofs of certain statements concerning the interplay between the game O F^ on subsets of W° and the filter games considered in [17] and [18].…”

Section: ) the Union A\ U A 2 Fails To Be Oand -Bounded For All Filtersmentioning

confidence: 99%

Filter games and pathological subgroups of a countable product of lines

Банах¹,

Nickolas²,

Sanchis³

2006

J. Aust. Math. Soc.

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To each filter & on co, a certain linear subalgebra A (^) of R™, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter &. For example, if & is a free ultrafilter, then A(^) is a Baire subalgebra of R" for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernandez, Robbie and Tkachenko); and if & x and ^2 are two free filters on a> that are not near coherent (such filters exist under Martin's Axiom), then A(&{) and A(J? 2 ) are two o-bounded and OF-undetermined subalgebras of U." whose product A(^\) x A(^i) is OF-determined and not o-bounded (this resolves a problem of Tkachenko). It is also shown that the statement that the product of two o-bounded subrings of R" is o-bounded is equivalent to the set-theoretic principle NCF (Near Coherence of Filters); this suggests that Tkachenko's question on the productivity of the class of o-bounded topological groups may be undecidable in ZFC.

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“…We have not included the game theoretic version of Ramsey properties which is indeed a very interesting approach. We refer the reader to the work of Laflamme [39,40].…”

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%