The tightness about sequential fans and combinatorial properties

Continuing [6], [8] and [16], we study the consequences of the weak FreeseNation property of (P(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 (P(ω), ⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models.

show abstract

“…The following variant of non(I ) for an ideal I over R was considered in the papers [5], [15], [20] and [21]:…”

Section: Cardinal Invariantsmentioning

confidence: 99%

On the weak Freese–Nation property of ?(ω)

Fuchino¹,

Geschke

Soukupe

2001

Arch. Math. Logic

View full text Add to dashboard Cite

show abstract

“…After reviewing some cardinal invariants introduced in [6] which are variants of the bounding number b and the shrinking number b * in [3], we study in Section 3 the effect of the combinatorial principles C s (κ), Ĉs (κ) and HP(κ) on the values of these cardinal invariants.…”

Section: Introductionmentioning

confidence: 99%

Coloring ordinals by reals

Brendle

Fuchino

2007

Fund. Math.

View full text Add to dashboard Cite

Abstract. We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ) for regular κ > ℵ 1 and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals.These principles are strengthenings of C s (κ) and F s (κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ 2 , ℵ 1 ) (hence also IP(ℵ 2 , ℵ 2 ) as well as HP(ℵ 2 )) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ 2 , ℵ 2 ) (hence also HP(ℵ 2 )) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2).Relations between these principles and their influence on the values of the variations b ↑ , b h , b * , do of the bounding number b are studied. One of the consequences of HP(κ) besides C s (κ) is that there is no projective wellordering of length κ on any subset of ω ω. We construct a model in which there is no projective well-ordering of length ω 2 on any subset of ω ω (do = ℵ 1 in our terminology) while b * = ℵ 2 (Theorem 6.4).

show abstract

Characters of ultrafilters and tightness of products of fans

Hušek

2007

Topology and its Applications

View full text Add to dashboard Cite

The tightness about sequential fans and combinatorial properties

Cited by 5 publications

References 5 publications

On the weak Freese–Nation property of ?(ω)

On the weak Freese–Nation property of ?(ω)

Coloring ordinals by reals

Characters of ultrafilters and tightness of products of fans

Contact Info

Product

Resources

About