  A set $A$ is said to split a finite set $B$ if exactly half the elements of
$B$ (up to rounding) are contained in $A$. We study the dual notions: (1)
splitting family, which is a collection of sets such that any subset of
$\{1,\ldots,k\}$ is split by a set in the family, and (2) splittable family,
which is a collection of sets such that there is a single set $A$ that splits
each set in the family.
  We study the minimum size of a splitting family on $\{1,\ldots,k\}$, as well
as the structure of splitting families of minimum size. We use a mixture of
computational and theoretical techniques. We additionally study the related
notions of $\mathord{\leq}4$-splitting families and $4$-splitting families, and
we provide lower bounds on the minimum size of such families.
  Next we investigate splittable families that are just on the edge of
unsplittability in several senses. First, we study splittable families that
have the fewest number of splitters. We give a complete characterization in the
case of two sets, and computational results in the case of three sets. Second,
we define a splitting game, and study splittable families for which a splitter
cannot be found under adversarial conditions.
