, Volume 38, Issue 2, pp 389-397
Date: 11 Dec 2007

The Empty Hexagon Theorem

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Abstract

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 Ci P 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 ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C1 P is minimal and C4 P 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.