  A pattern is called universal in another collection of sets, when every set
in the collection contains some linear and translated copy of the original
pattern. Paul Erd\H{o}s proposed a conjecture that no infinite set is universal
in the collection of sets with positive measure. This paper explores an
analogous problem in the topological setting. Instead of sets with positive
measure, we investigate the collection of dense $G_{\delta}$ sets and in the
collection of generic sets (dense $G_{\delta}$ and complement has Lebesgue
measure zero). We refer to such pattern as topologically universal and
generically universal respectively. It is easy to show that any countable set
is topologically universal, while any set containing an interior cannot be
topologically universal. In this paper, we will show that Cantor sets on
${\mathbb R}^d$ are not topologically universal and Cantor sets with positive
Newhouse thickness on ${\mathbb R}^1$ are not generically universal. This gives
a positive partial answer to a question by Svetic concerning the Erd\H{o}s
similarity problem on Cantor sets. Moreover, we also obtain a higher
dimensional generalization of the generic universality problem.
