Note on the 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

A Positive Fraction Erdos - Szekeres Theorem

We prove a fractional version of the Erdős–Szekeres theorem: for any k there is a constant ck > 0 such that any sufficiently large finite set X ⊂ R2 contains k subsets Y1, . . . ,Yk , each of size ≥ ck |X |, such that every set {y1, . . . , yk} with yi ∈ Yi is in convex position. The main tool is a lemma stating that any finite set X ⊂ Rd contains “large” subsets Y1, . . . ,Yk such that all set...

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...