Some variations on the splitting number

Variations on the splitting number s are examined by localizing the splitting property to finite sets. To be more precise, rather than considering families of subsets of the integers that have the property that every infinite set is split into two infinite sets by some member of the family a stronger property is considered: Whenever an subset of the integers is represented as the disjoint union of a family of finite sets one can ask that each of the finite sets is split into two non-empty pieces by some member of the family. It will be shown that restricting the size of the finite sets can result in distinguishable properties. In §2 some inequalities will be established, while in §3 the main consistency result will be proved.