Lindsay Mullen
Seminar Presentation #3
November 18, 2013
• Geometry
• Discrete Geometry
Discrete Geometry

Closely related to combinatorial geometry

Branches of geometry that study combinatorial properties
and constructive methods of discrete geometric objects

Most questions involve: finite or discrete sets of basic
geometric objects, such as points, lines, planes, circles,
spheres, polygons, and so forth

Focuses on the combinatorial properties of these objects,
such as how they intersect one another, or how they may be
arranged to cover a larger object
Definitions
 General
collinear
Position - no three points are
Definitions continued…

Convex Polygon - A polygon is convex if
there are no "dents" or indentations in it (no
internal angle is greater than 180°)
Definitions continued…

Convex hull of X - the
smallest convex set that
contains X. For instance,
when X is a bounded
subset of the plane, the
convex hull may be
visualized as the shape
formed by a rubber band
stretched around X
The Happy Ending Problem

The Happy Ending Problem states:
“Any set of five points in the plane in general
position has a subset of four points that form the
vertices of a convex quadrilateral.”
 Notation would be f(4)=5

This was one of the original results that led
to the development of Ramsey theory
History
1933 – Budapest, Hungary – a group of
students known as the Anonymous Society
would meet to discuss mathematics
 Group consisted of Paul Erdös, Esther Klein,
Endre Makai, and György (George)
Szekeres
 Klein posed the following question after
some doodling: “how many points do we
need on a plane to guarantee that some four
of them form a convex quadrilateral?”

History continued…
Erdös, Szekeres, and Klein sought to formalize
the result [f(4)=5] and develop a proof for the
case with more than 5 points
 Klein came up with a proof of different cases for
f(4)=5
 Szekeres and Erdös issued a paper, A
Combinatorial Problem in Geometry, in 1935
formally proving the result
 Specifically generalized a stronger theorem
known as the Erdös-Szekeres Conjecture:

 “For any positive integer N, any sufficiently large finite
set of points in the plane in general position has a
subset of N points that form the vertices of a convex
polygon.”
History continued…
Erdös dubbed this the Happy Ending Problem
because shortly after these results were
published, Szekeres and Klein got married
 Erdös and Szekeres (1935) claimed Endre
Makai proved that f(5)=9, or in words, every set
of nine points in the general position contains at
least one convex pentagon.
 HOWEVER, Makai never published a formal
proof so in 1970, J.D. Kalbfleish published a
proof for this result
 2006 – Szekeres and Lindsay Peters provided a
computer proof of f(6)=17, or 17 points in the
general position always contains a convex
hexagon.

Happy Ending Problem continued…

“Any set of five points in the plane in general
position has a subset of four points that form the
vertices of a convex quadrilateral.”
Klein’s Proof

Clearly, four points are not enough: with
some bad luck, three of them form a triangle
and the fourth lies inside the triangle formed
by the other three, and as a result, there are
no four points that form a convex
quadrilateral. So f(4)>4.

However, 5 points will work for any case!
Klein’s Cases
1.) If the five points lie on a pentagon, then any
four of them form a convex quadrilateral.
 2.) If four points form a convex quadrilateral
with the remaining point in the interior, then we
are done again, for the four “outer” points take
on the desired shape.
 3.) Finally, if three points form a triangle and
both the remaining points lie in the interior of
the triangle, then one of the sides of the triangle
together with the line that joins the two interior
points will form a convex quadrilateral.

Klein’s Cases
Case 1
Case 2
Case 3
What We Have So Far

So far the Happy Ending Problem has given
us the following:
 f(3) = 3, trivially.
 f(4) = 5
 f(5) = 9
 f(6) = 17
Current Research
The value of f(N) is unknown for all N > 6
 By the result of Erdös & Szekeres (1935) it
is known to be finite.
 Follows from the Erdös-Szekeres
Conjecture:

 “For any positive integer N, any sufficiently large
finite set of points in the plane in general position
has a subset of N points that form the vertices of
a convex polygon.”
While it remains unproven, there are less
precise bounds that are known
 the best known upper bound when N ≥ 7 is

References
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http://en.wikipedia.org/wiki/Discrete_geometry
http://planetmath.org/HappyEndingProblem
http://www.mathopenref.com/point.html
http://en.wikipedia.org/wiki/Convex_hull
http://www.mathsisfun.com/definitions/convex.html
http://neeldhara.com/ramblings/notes/cgt-01
http://en.wikipedia.org/wiki/Happy_Ending_problem
http://online.sfsu.edu/meredith/301/Papers/Happy%20Endi
ng%20Problem%20Term%20Paper%20(May%201%2020
11).pdf
http://cg.iit.bme.hu/~zsolnai/writings/ramsey_happy_end/ra
msey_happy_end.pdf
http://sshanshans.wordpress.com/2011/12/01/the-happyending-problem-2/