A Quantitative Doignon-Bell-Scarf theorem

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and on the number k, such that if a bounded polyhedron {x in R^n : Ax <= b} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n,k), defining a polyhedron that contains exactly the same k integer points.
In this case c(n,0)=2^n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities.
We work on both upper and lower bounds for the constant c(n,k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function.