The Relative Consistency Of

We prove the consistency result from the title. By forcing we construct a model of g = ℵ1, b = cf(Sym(ω)) = ℵ2. We recall the definitions of the three cardinal characteristics in the title and the abstract. We write A ⊆ * B if A \ B is finite. We write f ≤ * g if f, g ∈ ω ω and {n : f (n) > g(n)} is finite. Definition 0.1. (1) A subset G of [ω] ω is called groupwise dense if – for all B ∈ G, A ⊆ * B we have that A ∈ G and – for every partition {[π i , π i+1) : i ∈ ω} of ω into finite intervals there is an infinite set A such that {[π i , π i+1) : i ∈ A} ∈ G. The groupwise density number, g, is the smallest number of groupwise dense families with empty intersection. (2) Sym(ω) is the group of all permutations of ω. If Sym(ω) = i<κ K i and κ = cf(κ) > ℵ 0 , K i : i < κ is increasing and continuous, K i is a proper subgroup of Sym(ω), we call K i : i < κ a cofinality witness. We call the minimal such κ the cofinality of the symmetric group, short cf(Sym(ω)). (3) The bounding number b is b = min{|F| : F ⊆ ω ω ∧ (∀g ∈ ω ω)(∃f ∈ F)f ≤ * g}. Simon Thomas asked whether g = cf(Sym(ω)) is consistent [8, Question 3.1]. In this paper we prove: Theorem 0.2. g < cf(Sym(ω)) is consistent relative to ZFC.