Higher-order Erdos-Szekeres theorems

Citations of this article
Mendeley users who have this article in their library.


Let P = (p1, p2,..., p N) be a sequence of points in the plane, where p i = (x i, y i) and x1 < x2 < ⋯ < x N. A famous 1935 Erdos-Szekeres theorem asserts that every such P contains a monotone subsequence S of ⌈N⌉ points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Ω (log N) points. Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k + 1) -tuple K ⊆ P to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative(k + 1) -tuple. Then we say that S ⊆ P is k th-order monotone if its (k + 1) -tuples are all positive or all negative. We investigate a quantitative bound for the corresponding Ramsey-type result (i.e., how large kth-order monotone subsequence can be guaranteed in every N-point P). We obtain an Ω (log (k -1)N) lower bound ((k - 1) -times iterated logarithm). This is based on a quantitative Ramsey-type theorem for transitive colorings of the complete (k + 1) -uniform hypergraph (these were recently considered by Pach, Fox, Sudakov, and Suk). For k = 3, we construct a geometric example providing an O (log log N) upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for order-type homogeneous subsets in R3, as well as for a Ramsey-type theorem for hyperplanes in R4 recently used by Dujmović and Langerman. © 2013 Elsevier Ltd.




Eliáš, M., & Matoušek, J. (2013). Higher-order Erdos-Szekeres theorems. Advances in Mathematics, 244, 1–15. https://doi.org/10.1016/j.aim.2013.04.020

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free