Bounded VC-Dimension Implies a Fractional Helly Theorem

Abstract

We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system $\FF$ is bounded by $o(m^k)$, then $\FF$ has fractional Helly number $k$. This means that for every $\alpha>0$ there exists a $\beta>0$ such that if $F_1,F_2,\ldots,F_n\in\FF$ are sets with $\bigcap_{i\in I}F_i\neq\emptyset$ for at least $\alpha{n\choose k}$ sets $I\subseteq\{1,2,\ldots,n\}$ of size $k$, then there exists a point common to at least $\beta n$ of the $F_i$. This further implies a $(p,k)$-theorem: for every $\FF$ as above and every $p\geq k$ there exists $T$ such that if $\GG\subseteq\FF$ is a finite subfamily where among every $p$ sets, some $k$ intersect, then $\GG$ has a transversal of size $T$. The assumption about bounded dual shatter function applies, for example, to families of sets in $\Rd$ definable by a bounded number of polynomial inequalities of bounded degree; in this case we obtain fractional Helly number $d{+}1$.