Problem 2

Does there exist a finite set

M

of points in space, not all in one plane, such that for every two points

A,B\in M

there are two other points

C,D\in M

for which the lines

AB

and

CD

are parallel but distinct?

Step 1 of 5: Construct the set

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In plain words

Two perpendicular planes through one diameter make the union genuinely three-dimensional.

H_1,H_2\text{ share a diameter and have perpendicular planes}

Detailed analysis

Take two congruent regular hexagons with the same center and one common diameter, in two distinct planes perpendicular along that diameter. Let $M=H_1\cup H_2$ .