On splitting and splittable families
Abstract
A set $A$ is said to \emph{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) a \emph{splitting family}, a collection of sets such that any subset of $\{1,\ldots,k\}$ is split by a set in the family, and (2) a \emph{splittable family}, 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 the \emph{splitting game}, and study splittable families for which a splitter cannot be found under adversarial conditions.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.12896
 Bibcode:
 2018arXiv181112896C
 Keywords:

 Mathematics  Combinatorics;
 05A18;
 05D05;
 11K38