We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\rightarrow [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which f is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* <\mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $\kappa $ such that any two disjoint countable dense sets in $2^{\mathbb {N}}$ can be separated by sets each of which is an intersection of at most $\kappa $-many open sets in $2^{\mathbb {N}}$. We establish also that $\mathfrak d^*=\min \{\mathfrak u, \mathfrak d\}=\min \{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.

Minimax estimation procedures for the mean vector of a distribution on a compact set under squared error type loss functions are considered. In particular, a Dirichlet process prior is used to show that a linear function of is a minimax estimator in the class of all measurable estimators and all possible distributions. This effort extends some earlier work of Bühlmann to a more general setting.