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Combinatorics of dense subsets of the rationals

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Balcar B., Hernández-Hernández F., Hrušák M.

Fundamenta Mathematicae

2004

tom 183

nr 1

59-80

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $𝔭_{ℚ}$, $𝔱_{ℚ}$, $𝔥_{ℚ}$, $𝔰_{ℚ}$, $𝔯_{ℚ}$, $𝔦_{ℚ}$ describing properties of Dense(ℚ). These invariants satisfy $𝔭_{ℚ}$ ≤ 𝔱_{ℚ} ≤ 𝔥_{ℚ} ≤ 𝔰_{ℚ} ≤ 𝔯_{ℚ} ≤ 𝔦_{ℚ}$. We compare them with their analogues in the well studied Boolean algebra 𝒫(ω)/fin. We show that $𝔭_{ℚ} = p$, $𝔱_{ℚ} = t$ and $𝔦_{ℚ} = i$, whereas $𝔥_{ℚ} > h$ and $𝔯_{ℚ} > r$ are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of ,ℚ. In particular, we show that non(M)=min{|𝒟|: 𝒟 ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ 𝒟)(I ∩ D = ∅)} and cof(M) = min{|𝒟|: 𝒟 ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ 𝒟)(I ∩ 𝒟 = ∅)}. We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

Incomparable families and maximal trees

100%

Campero-Arena G., Cancino J., Hrušák M., Miranda-Perea F.

Fundamenta Mathematicae

2016

tom 234

nr 1

73-89

We answer several questions of D. Monk by showing that every maximal family of pairwise incomparable elements of 𝒫(ω)/fin has size continuum, while it is consistent with the negation of the Continuum Hypothesis that there are maximal subtrees of both 𝒫(ω) and 𝒫(ω)/fin of size ω₁.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Hrušák M.

1 Balcar B.

1 Campero-Arena G.

1 Cancino J.

1 Hernández-Hernández F.

1 Miranda-Perea F.

1 2016

1 2004

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Combinatorics of dense subsets of the rationals

Incomparable families and maximal trees