  We provide a new characterization of quasi-analyticity of Denjoy-Carleman
classes, related to \emph{Wetzel's Problem}. We also completely resolve which
Denjoy-Carleman classes carry \emph{sparse systems}: if the Continuum
Hypothesis (CH) holds, \textbf{all} Denjoy-Carleman classes carry sparse
systems; but if CH fails, a Denjoy-Carleman class carries a sparse system if
and only if it is not quasi-analytic. As corollaries, we extend results of
\cite{MR3552748} and \cite{CodyCoxLee} about non-existence of "anonymous
predictors" for real functions.
