On the Computational Complexity of Erdős-Szekeres and Related Problems in ℝ3

The Erdős-Szekeres theorem states that, for every k, there is a number n k such that every set of n k points in general position in the plane contains a subset of k points in convex position. If we ask the same question for subsets whose convex hull does not contain any other point from the set, this is not true: as shown by Horton, there are sets of arbitrary size that do not contain an empty convex 7-gon.