A Positive Fraction Erdos - Szekeres Theorem

Erdos-Szekeres theorem with forbidden order types

According to the classical Erdős–Szekeres theorem, every sufficiently large set of points in general position in the plane contains a large subset in convex position. Parallel to the Erdős–Hajnal problem in graph-Ramsey theory, we investigate how large such subsets must a configuration contain if it does not have any sub-configuration belonging to a fixed order type. © 2005 Elsevier Inc. All ri...

A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets

Note on the Erdos - Szekeres Theorem

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935 P. Erd} os and G. Szekeres showed that g(n) exists and 2 n?2 + 1 g(n) 2n?4 n?2 + 1. Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this note we further improve the upper bound to

Combinatorial Problems in Geometry and Number Theory

In a previous paper (given at the International Congress of Mathematicians at Nice 1970) entitled "On the application of combinatorial analysis to number theory geometry and analysis", I discussed many combinatorial results and their applications . I will refer to this paper as I (as much as possible I will try to avoid overlap with this paper) . First I discuss those problems mentioned in I wh...

Two player game variant of the Erdos-Szekeres problem

The classical Erdős-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdős-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k...