Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg

We will show that the following set theoretical assumption \continuum=\omega2, the dominating number d equals to \omega1, and there exists an \omega1-generated Ramsey ultrafilter on \omega (which is consistent with ZFC) implies that for an arbitrary sequence fn:R--\u3eR of uniformly bounded functions there is a subset P of R of cardinality continuum and an infinite subset W of \omega such that {fn|P: n in 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(null)=\omega1 implies existence of a magic set and of a function f:R--\u3eR such that f|D is discontinuous for every D which is not simultaneously meager and of measure zero