Combinatorics of dense subsets of the rationals

Abstract

We study combinatorial properties of the partial order (Dense(ℚ), ⊆. To do that we introduce cardinal invariants pℚ, t ℚ, hℚ, sℚ, tℚ, iℚ describing properties of Dense(ℚ). These invariants satisfy pℚ ≤ tℚ ≤ hℚ ≤ sℚ ≤ tℚ ≤ iℚ. We compare them with their analogues in the well studied Boolean algebra P(ω)/fin. We show that pℚ = p, tℚ = t and i ℚ = i, whereas hℚ > h and tℚ > t 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 = 0φ)} and cof (M) = min{|D| : D ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ D)(I ∩ D = 0φ)}. We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.