Abstract
We will show that the following set theoretical assumption
$\mathfrak{c}=\omega_2$, the dominating number $\mathfrak {d}$ equals to $\omega_1$, and there exists an $\omega_1$-generated Ramsey ultrafilter on $\omega$
(which is consistent with ZFC) implies that for an arbitrary sequence $f_n\colon\mathbb{R}\to\mathbb{R}$ of uniformly bounded functions there is a set $P\subset\mathbb{R}$ of cardinality continuum and an infinite $W\subset\omega$ such that $\{f_n\restriction P\colon n\in W\}$ is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions $f_n$ are measurable or have the Baire property then $P$ can be chosen as a perfect set. We will also show that cof$(\mathcal{N})=\omega_1$ implies existence of a magic set and of a function $f\colon\mathbb{R}\to\mathbb{r}$ such that $f\restriction D$ is discontinuous for every $D\notin\mathcal{N}\cap\mathcal{M}$.
Citation
Krzysztof Ciesielski. Janusz Pawlikowski. "Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg." Real Anal. Exchange 26 (2) 905 - 912, 2000/2001.
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