If two lattice points in the plane are chosen and joined by a line segment, the midpoint of this segment may or may not also be a lattice point.

Suppose you try selecting a set of points so that for every pair, the midpoint is not a lattice point.

Is there a limit to how many points you can pick?

(In reply to

spoiler by xdog)

huh? Ady is right. No sqrt(2) is involved. The midpoint of (a,b) and (c,d) is ((a+c)/2,(b+d)/2). It is a lattice point if and only if a and c are of the same parity, and b and d are also of the same parity. And thus the set of points are limited to 4.