I'm looking for the least possible value of k such that we can always choose 2 points out of k points in n-dimensional Euclidean space such that there's at least one more lattice point on the segment joining them. Since this might not be terribly clear, let me give an example. If you have 9 distinct lattice points in three-dimensional Euclidean space, you can show that there is a lattice point on the interior of one of the line segments joining two of these points. I am looking to generalize this to the n-dimensional case. Thanks.
Conditions:
In n-dimensional space, every point can be represented as an n-tuple of real numbers.
Lattice points are defined as points that have integer coordinates.
If two points are congruent modulo $2$ then their midpoint will be a lattice point. To force this you need $2^n+1$ points.
But with $2^n$ points you have the vertices of the unit cube (all vectors with zero/one entries) and no pair of these has a lattice point in between. So $2^n+1$ is optimal.