Using partitions to characterize the minimum cardinality of an unbounded family in ωω

AbstractLet b be the minimum cardinality of an unbounded family in ωω partially ordered by ⩽∗. Recall that ƒ ⩽ ∗ g if ƒ(n)⩽g(n) for all but a finite number of n We first generalize ⩽∗ to νω for each infinite cardinal ν. We then prove that b > ν iff every sequence in νω has a ⩽ ∗-upper bound; we also show that the existence of an upper bound for a sequence σ, where σk: ν → ω for each k ϵ ω, depends only on the partitions of ν induced by the point-inverse sets of the individual functions