SMALL COMBINATORIAL CARDINAL CHARACTERISTICS AND THEOREMS OF EGOROV AND BLUMBERG

We will show that the following set theoretical assumption c = ω2, the dominating number d equals to ω1, and there exists an ω1-generated Ramsey ultrafilter on ω (which is consistent with ZFC) implies that for an arbitrary sequence fn : R → R of uniformly bounded functions there is a set P ⊂ R of cardinality continuum and an infinite W ⊂ ω such that {fn P : n ∈ W} is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions fn are measurable or have the Baire property then P can be chosen as a perfect set. We will also show that cof(N ) = ω1 implies existence of a magic set and of a function f : R → R such that f D is discontinuous for every D / ∈ N ∩M. Our set theoretic terminology is standard and follows that of [8]. In particular, |X| stands for the cardinality of a set X and c = |R|. We are using