Combinatorics (Fall 2010)/Ramsey theory: Difference between revisions
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[http://www.maa.org/mathland/mathtrek_10_3_00.html] for the proof. | [http://www.maa.org/mathland/mathtrek_10_3_00.html] for the proof. | ||
We say | We say a set of points in the plane in [http://en.wikipedia.org/wiki/General_position '''general positions'''] if no three of the points are on the same line. | ||
{{Theorem|Theorem (Erdős-Szekeres 1935)| | {{Theorem|Theorem (Erdős-Szekeres 1935)| |
Revision as of 08:58, 17 November 2010
Ramsey's Theorem
Ramsey number
The "Happy Ending" problem
The happy ending problem - Any set of 5 points in the plane, no three on a line, has a subset of 4 points that form the vertices of a convex quadrilateral.
See the article [1] for the proof.
We say a set of points in the plane in general positions if no three of the points are on the same line.
Theorem (Erdős-Szekeres 1935) - For any positive integer [math]\displaystyle{ n\ge 3 }[/math], there is an [math]\displaystyle{ N(n) }[/math] such that any set of at least [math]\displaystyle{ N(n) }[/math] points in general position in the plane (i.e., no three of the points are on a line) contains [math]\displaystyle{ n }[/math] points that are the vertices of a convex [math]\displaystyle{ n }[/math]-gon.