A fractional Helly theorem for convex lattice sets

Abstract

A set of the form C∩Zd, where C⊆Rd is convex and Zd denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2d. We prove that the fractional Helly number is only d+1: For every d and every α∈(0,1] there exists β>0 such that whenever F1,…,Fn are convex lattice sets in Zd such that ⋂i∈IFi≠∅ for at least α(nd+1) index sets I⊆{1,2,…,n} of size d+1, then there exists a (lattice) point common to at least βn of the Fi. This implies a (p,d+1)-theorem for every p⩾d+1; that is, if F is a finite family of convex lattice sets in Zd such that among every p sets of F, some d+1 intersect, then F has a transversal of size bounded by a function of d and p.