Bounded VC-Dimension Implies a Fractional Helly Theorem

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 F is bounded by o(m k), then F has fractional Helly number k. This means that for every > 0 there exists a > 0 such that if F 1 ; F 2 ; : : : ; F n 2 F are sets with T i2I F i 6 = ; for at least ? n k sets I f1; 2; : : :; ng of size k, then there exists a point common to at least n of the F i. This further implies a (p; k)-theorem: for every F as above and every p k there exists T such that if G F is a nite subfamily where among every p sets, some k intersect, then G has a transversal of size T. The assumption about bounded dual shatter function applies, for example, to families of sets in R d deenable by a bounded number of polynomial inequalities of bounded degree; in this case, we obtain fractional Helly number d+1.