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A NOTE ON ORDER-TYPE HOMOGENEOUS POINT SETS

Part of: Extremal combinatorics Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  19 December 2013

Andrew Suk
Andrew Suk*
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA,U.S.A. email asuk@math.mit.edu
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Abstract

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

MSC classification

Secondary: 05D10: Ramsey theory 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52B11: $n$-dimensional polytopes
Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 37 - 42
Copyright
Copyright © University College London 2013 

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