A fractional Helly theorem for convex lattice sets

A set of the form C-Z ; where CDR is convex and Z denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2 : We prove that the fractional Helly number is only d þ 1: For every d and every aAð0; 1 there exists b40 such that whenever F1;y;Fn are convex lattice sets in Z such that T iAI Fia| for at least að n dþ1Þ index sets IDf1; 2;y; ng of size d þ 1; then there exists a (lattice) point common to at least bn of the Fi: This implies a ðp; d þ 1Þ-theorem for every pXd þ 1; that is, ifF is a finite family of convex lattice sets in Z such that among every p sets of F; some d þ 1 intersect, then F has a transversal of size bounded by a function of d and p: r 2003 Elsevier Science (USA). All rights reserved.