Erdős-Szekeres-type theorems for monotone paths and convex bodies

Abstract

For any sequence of positive integers j 1 < j 2 < · · · < j n , the k-tuples (j i , j i+1 , ..., j i+k−1 ), i = 1, 2, . . . , n−k+1, are said to form a monotone path of length n. Given any integers n ≥ k ≥ 2 and q ≥ 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N ] = {1, 2, . . . , N } with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N k (q, n), it follows from the seminal 1935 paper of Erdős and Szekeres that N 2 (q, n) = (n − 1) q + 1 and N 3 (2, n) = 2n−4 n−2 + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2for q ≥ 2 and n ≥ q + 2. Using a "stepping-up" approach that goes back to Erdős and Hajnal, we prove analogous bounds on N k (q, n) for larger values of k, which are towers of height k − 1 in n q−1 . As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M (n) = 2 n 2 log n plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union.

Three-Monotone Interpolation

“…Here we consider this question but in a slightly different and technically more convenient formulation. (Let us remark that a number of other generalizations of the Erdős-Szekeres theorems have recently been considered [6,7,9,11,23]. )…”

Increasing Hamiltonian paths in random edge orderings

“…, n 2 + 1} contains a monotonic subsequence of length n + 1. Many extensions have been found for this theorem: see, e.g., any of [7,12,14,16,17]. In this paper, we Correspondence to: Mikhail Lavrov *Supported by NSF grant (DMS-1201380); NSA Young Investigators Grant; USA-Israel BSF Grant.…”

“…The classical result of Erdős and Szekeres states that any permutation of contains a monotonic subsequence of length n + 1. Many extensions have been found for this theorem: see, e.g., any of . In this paper, we consider the direction started by Chvátal and Komlós .…”

Ordered Ramsey theory and track representations of graphs

“…Focusing on ordered paths, Fox, Pach, Sudakov, and Suk [9] defined N k (t, m) to be the least integer N such that in every t-coloring of the k-subsets of N linearly ordered elements there is a monochromatic k-uniform ordered path with m edges. That is, N k (t, m) = OR t (P k m+k−1 ) in our notation.…”

“…. , H t , the problem is particularly fundamental for ordered paths, because this case has applications (see [9]) and reduces to computing the size of a natural poset. (Choudum and Ponnusamy [3] used "ordered Ramsey number" for a different concept involving colored tournaments.…”