Sách Another abstraction of the Erd˝os-Szekeres Happy End Theorem

Abstract
The Happy End Theorem of Erd˝os 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.
hen she shared this news with a circle of her friends in Budapest, the following prospect
generalizing it emerged:
Can we find for each integer n greater than two an integer N(n) such that
from any set of N(n) points in the plane of which no three lie on the same line
it is always possible to select n points that are vertices of a convex polygon?
ndre Makai proved that N(5) = 9 works here. A few weeks later, George Szekeres proved
e existence of N(n) for all n. His argument produced very large upper bounds for N(n):
r instance, it gave N(5) 6 2
10000
. Soon afterwards, Paul Erd˝os came up with a different
oof, which led to much smaller values of N(n):