On General Boundedness and Dominating Cardinals

For cardinals κ, λ,μ we let bκ,λ,μ be the smallest size of a subset B of λμ unbounded in the sense of ≤κ ; that is, such that there is no function f ∈ λμ such that {α < λ : g(α) > f (α)} has size less than κ for all g ∈ B. Similarly for dκ,λ,μ, the general dominating number, which is the smallest size of a subset B of λμ such that for every g ∈ λμ there is an f ∈ B such that the above set has size less than κ . These cardinals are generalizations of the usual ones for κ = λ = μ = ω. When all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version b ↑ κ,λ,μ of bκ,λ,μ in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) bμ,μ,cfμ ≤ bcfμ,cfμ,cfμ; (2) for λ ≤ μ, b ↑ κ,λ,μ always exists; (3) if cfλ = cfμ < λ ≤ μ, then bcfμ,cfμ,cfμ = b ↑ λ,λ,μ; (4) dω,μ,μ = d1,μ,μ. For background see Section 1 of the paper. Several open problems are stated. 1 Definitions We make the standing assumptions that we have cardinals κ, λ,μ with (1) κ = 1 or κ is infinite, (2) κ ≤ λ, and (3) λ and μ are infinite. Note in particular that we allow for the possibility that λ > μ. The definitions of our functions depend on some quasi orders defined as follows. For f, g ∈ μ we write f ≤κ g iff |{ξ < λ : f (ξ) > g(ξ)}| < κ, f <κ g iff |{ξ < λ : f (ξ) ≥ g(ξ)}| < κ, f ≤ g iff ∀ξ < λ[ f (ξ) ≤ g(ξ)], f < g iff ∀ξ < λ[ f (ξ) < g(ξ)]. The following obvious proposition can be used to fill in some details below. Received April 23, 2003; accepted February 5, 2004; printed October 26, 2004 2000 Mathematics Subject Classification: Primary, 03E10; Secondary, 03E35