Official solution came out today:
SOLUTION: The first issue to be addressed here is: When do two lattice points, say (a,b,c) and (d,e,f), have another lattice point on the line segment between them? A moment's thought will convince you that this happens when the numbers a - d, b - e, and c - f have a common divisor. The "easiest" common divisor for them to have is 2, and this will happen if a - d, b - e, and c - f are all even, i.e., if the two lattice points have the same parity coordinatewise. When that happens, the midpoint of the two lattice points, namely, the vector ((a + d)/2, (b + e)/2, (c + f)/2), will be a lattice point.
But there are only 2^3 = 8 different parities available, so by the pigeonhole principle, we can't have more than eight points in our set L. Can we have eight? Yes, just let L consist of the corners of a unit cube
