The empty hexagon theorem

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let C-P(i) be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P boolean AND C(S) different from S has cardinality strictly less than vertical bar S vertical bar. Our main theorem states that P contains an empty convex hexagon if C-P(1) is minimal and C-P(4) is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.