  We introduce the problem Partial VC Dimension that asks, given a hypergraph
$H=(X,E)$ and integers $k$ and $\ell$, whether one can select a set $C\subseteq
X$ of $k$ vertices of $H$ such that the set $\{e\cap C, e\in E\}$ of distinct
hyperedge-intersections with $C$ has size at least $\ell$. The sets $e\cap C$
define equivalence classes over $E$. Partial VC Dimension is a generalization
of VC Dimension, which corresponds to the case $\ell=2^k$, and of
Distinguishing Transversal, which corresponds to the case $\ell=|E|$ (the
latter is also known as Test Cover in the dual hypergraph). We also introduce
the associated fixed-cardinality maximization problem Max Partial VC Dimension
that aims at maximizing the number of equivalence classes induced by a solution
set of $k$ vertices. We study the algorithmic complexity of Partial VC
Dimension and Max Partial VC Dimension both on general hypergraphs and on more
restricted instances, in particular, neighborhood hypergraphs of graphs.
