Tight bounds on discrete quantitative Helly numbers

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.