Forced Convex n -Gons in the Plane

In a seminal paper from 1935, Erd}os and Szekeres showed that for each n there exists a least value g(n) such that any subset of g(n) points in the plane in general position must always contain the vertices of a convex n-gon. In particular, they obtained the bounds 2n 2 + 1 g(n) 2n 4 n 2 + 1 ; which have stood unchanged since then. In this note we remove the +1 from the upper bound for n 4. 1. The main result In 1935, Paul Erd} os and George Szekeres published a short paper \A combinatorial problem in geometry" [1] which was destined to have a profound in uence on the development of combinatorics (and especially Ramsey theory) during the next 60 years (cf. [3]). In particular, in this paper Erd}os and Szekeres rediscovered Ramsey's theorem, which had only just appeared (unknown to them) ve years earlier. Their investigations arose from a geometrical question of the talented young mathematician Esther Klein (soon to become Mrs. Szekeres). She asked, \Is it true that for every n, there is a least value g(n) such that any set X of g(n) points in the plane in general position always contains the vertices of a convex n-gon?" Erd}os and Szekeres gave several proofs of the existence of g(n) in [1] and established the following bounds: 2 2 + 1 g(n) 2n 4 n 2 ! + 1 : (1) They also conjectured that the lower bound in (1) in fact always holds with equality. This is known [2] to be the case for n 5. Despite repeated attempts over the years, no general improvement on (1) has been found. In this note, we make a very small improvement on the upper bound of (1). Namely, we show g(n) 2n 4 n 2 ! (2)