  The famous Doignon-Bell-Scarf Theorem is a Helly-type result about the
existence of integer solutions on 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 $k$, such that if a polyhedron ${x: Ax \leq b}$ contains
exactly k integer solutions, 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 $c(n,0) = 2^n$ is 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.
