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 F is bounded by o(m ), then F has fractional Helly number k. This means that for every ff ? 0 there exists a fi ? 0 such that if F 1 ; F 2 ; : : : ; Fn 2 F are sets with i2I F i 6= ; for at least sets I ` f1; 2; : : : ; ng of size k, then there exists a point common to at least fin 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 finite 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 definable by a bounded number of polynomial inequalities of bounded degree; in this case, we obtain fractional Helly number d+1.

Citations