Can groupwise density be much bigger than the non-dominating number.pdfVIP

a r X i v : m a t h / 0 6 1 2 3 5 3 v 2 [ m a t h .L O ] 1 5 A u g 2 0 0 7 GROUPWISE DENSITY CANNOT BE MUCH BIGGER THAN THE UNBOUNDED NUMBER SAHARON SHELAH Abstract. We prove that g (the groupwise density number) is smaller or equal to b+, the successor of the minimal cardinality of an unbounded subset of ωω. 1. Introduction In the present note we are interested in two cardinal characteristics of the con- tinuum, the unbounded number b and the groupwise density number g. The former cardinal belongs to the oldest and most studied cardinal invariants of the contin- uum (see, e.g., van Douwen [9] and Bartoszyn?ski and Judah [2]) and it is defined as follows. Definition 1.1. (a) The partial order ≤Jbdω on ωω is defined by f≤Jbdω g if and only if (?N ω)(?n N)(f(n) ≤ g(n)). (b) The unbounded number b is defined by b = min{|F| : F ? ωω has no ≤Jbdω –upper bound in ωω}. The groupwise density number g, introduced in Blass and Laflamme [4], is per- haps less popular but it has gained substantial importance in the realm of cardi- nal invariants. For instance, it has been studied in connection with the cofinality c(Sym(ω)) of the symmetric group on the set ω of all integers, see Thomas [8] or Brendle and Losada [5]. The cardinal g is defined as follows. Definition 1.2. (a) We say that a family A ? [ω]?0 is groupwise dense when- ever: ? B ? A ∈ A, B ∈ [ω]?0 implies B ∈ A, and ? for every increasing sequence 〈mi : i ω〉 ∈ ωω there is an infinite set U ? ω such that ? {[mi,mi+1) : i ∈ U} ∈ A. (b) The groupwise density number g is defined as the minimal cardinal θ for which there is a sequence 〈Aα : α θ〉 of groupwise dense subsets of [ω]?0 such that ( ?B ∈ [ω]?0 )( ?α θ )( ?A ∈ Aα )( B 6?? A ) . (Recall that for infinite sets A and B, A ?? B means A \B is finite.) The unbounded number b and groupwise density number g can be in either order, see Blass [3] and Mildenberger and Shelah [7], [6]. However, as we show in Theorem 2.2, g cannot be bigger than b+. Date: August 20