Crossing segments and their parity among a point set

From any 5 points in the plane one can choose two disjoint pairs of points such that the segment joining the points of the first pair intersects the segment joining the points of the second pair.

The Conway–Gordon–Sachs theorem for linear embeddings. For any 6 points in general position in space there exist two linked triangles with vertices at these points (i.e., such that the union of the sides of the first one intersects the second two-dimensional triangle in exactly one point.) (A set of points in space is said to be in general position if no 4 of them lie in one plane.)

The van Kampen–Flores theorem for linear embeddings. From any 7 points in four-dimensional space one can choose two disjoint triples of points such that the two-dimensional triangles formed by these triples intersect.