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Summary: Another abstraction of the Erdos-Szekeres
Happy End Theorem
Noga Alon
Ehsan Chiniforooshan
Vasek Chv´atal
Fran¸cois Genest §
Submitted: July 13, 2009; Accepted: Jan 26, 2010; Published: XX
Mathematics Subject Classification: 05D10
Abstract
The Happy End Theorem of Erdos and Szekeres asserts that for every integer n
greater than two there is an integer N such that every set of N points in general
position in the plane includes the n vertices of a convex n-gon. We generalize this
theorem in the framework of certain simple structures, which we call "happy end
spaces".
In the winter of 1932/33, Esther Klein observed that
from any set of five points in the plane of which no three lie on the same line
it is always possible to select four points that are vertices of a convex polygon.
When she shared this news with a circle of her friends in Budapest, the following prospect
of generalizing it emerged:
Can we find for each integer n greater than two an integer N(n) such that
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