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Combinatorics of dense subsets of the rationals

B. BalcarF. Hernández-HernándezM. Hrušák — 2004

Fundamenta Mathematicae

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants , , , , , describing properties of Dense(ℚ). These invariants satisfy ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ . W e c o m p a r e t h e m w i t h t h e i r a n a l o g u e s i n t h e w e l l s t u d i e d B o o l e a n a l g e b r a ( ω ) / f i n . W e s h o w t h a t ℚ = p , ℚ = t a n d ℚ = i , w h e r e a s ℚ > h a n d ℚ > r a r e b o t h s h o w n t o b e r e l a t i v e l y c o n s i s t e n t w i t h Z F C . W e a l s o i n v e s t i g a t e c o m b i n a t o r i c s o f t h e i d e a l n w d o f n o w h e r e d e n s e s u b s e t s o f , . I n p a r t i c u l a r , w e s h o w t h a t non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

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