Erdos-Szekeres-type theorems for monotone paths and convex bodies
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Fox_Erdos-Szekeres.pdf
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Author(s)
Fox, Jacob
Pach, Janos
Sudakov, Benny
Suk, Andrew
Date Issued
November 2012
Journal
Proceedings of the London Mathematical Society
Publisher
Oxford University Press
Citation
Fox, Jacob et al. "Erdos-Szekeres-type theorems for monotone paths and convex bodies." Proceedings of the London Mathematical Society 105, 5 (November 2012): 953–982.
Version
Author's final manuscript
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\ge k\ge 2 and q\ge 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 Erd\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq q+2. Using a stepping-up approach that goes back to Erdos 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.
Description
Dedicated to the 75th anniversary of the publication of the Happy Ending Theorem
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://academic.oup.com/plms/article/105/5/953/1438687?login=true
