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?

I agree completely with respective well explained conclusions made by AT and SH that the required number of lattice points is indeed limited to 4.

*Edited on ***July 16, 2022, 3:22 am**