| I consider the problem of extending certain invariants of subsets of natural numbers to their equivalents for subsets of larger cardinals. The two specific questions I will address are the extension of the off-branch number o and its relation to other invariants; and the effect of replacing the ideal of sets of size < kappa with the ideal of non-stationary sets in cardinal invariants on kappa, with particular attention to the splitting number s (kappa).;The off-branch number on kappa is o (kappa), the least number of off-branch subsets of 2<kappa which together with the branches of 2kappa form a mad family on 2<kappa. The ZFC-provable inequalities I show are that a (kappa) ≤ o (kappa) and nonM (kappa) ≤ o (kappa) for kappa inaccessible. The consistency results I find are that a (kappa) < o (kappa) and nonM (kappa) < o (kappa) are possible if kappa is inaccessible, and that o (kappa) < 2kappa is possible if kappa is indestructibly weakly compact.;For an ideal I on kappa, the I -splitting number is sIk , the least size of a family S of sets in I+ such that for every set X ∈ I+ there is a set S ∈ S with X ⋂ S, X -- S ∈ I+ . For I the ideal of sets of size < kappa, this is the usual splitting number of kappa, whose being large has been shown to be equivalent to large cardinal properties; I obtain similar results for I = NS(kappa), the ideal of non-stationary subsets of kappa. |
