Recent progress on the Erdős-Szekeres conjecture

Martin Balko a Josef Cibulka

Abstract

In 1935, Erdős and Szekeres showed that for every integer k >= 2 there is a least number ES(k) <= 4^(k-o(k)) such that every set of ES(k) points in the plane in general position contains k points in convex position. They also posed the Erdős-Szekeres conjecture, which says that ES(k) = 2^(k-2)+1 for every integer k >= 2. Later, they supported this conjecture with the lower bound ES(k) >= 2^(k-2)+1.

Despite several efforts over the last 81 years, this conjecture remains open and, until very recently, no improvement in the order of magnitude has been made on the upper bound for ES(k).

In this talk we survey improvements on the upper bounds for ES(k) and we present a recent breakthrough result by Andrew Suk, who nearly settled the Erdős-Szekeres conjecture by showing that ES(k) <= 2^(k+o(k)). His result is based on an earlier work of A. Pór and P. Valtr and the proof is surprisingly