Tight bounds on discrete quantitative Helly numbers

Let S be a discrete subset of Rⁿ and define c(S,k) as the smallest number with the property that if a finite family of convex sets has exactly k points of S in common, then at most c(S,k) convex sets in this family already have exactly k points of S in common. For S=Zⁿ, this number repeatedly appeared in different contexts as, for instance, optimization and geometry of numbers and, very recently, for general sets S, in the context of Helly and Tverberg theorems in De Loera et al. (2015). In this work, we give a useful description of c(S,k) in terms of polytopes with vertices in S. Starting with this description, we answer several fundamental questions about c(S,k). We provide the general upper bound c(S,k)≤⌊(k+1)/2⌋(c(S,0)−2)+c(S,0) for every discrete S. For the integer lattice S=Zⁿ, employing techniques from the geometry of numbers, we solve the question on the asymptotic behavior by proving the estimate c(Zⁿ,k)=Θ(k^{(n−1)/(n+1)}) for every fixed n, and we compute the exact values of c(Zⁿ,k) for k=0,…,4.