Combinatorics (Fall 2010)/Ramsey theory: Difference between revisions
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=== The "Happy Ending" problem === | === The "Happy Ending" problem === | ||
{{Theorem|The happy ending problem| | {{Theorem|The happy ending problem| | ||
:Any set of 5 points in the plane | :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. | ||
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Revision as of 08:17, 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.
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 collection of [math]\displaystyle{ N\ge N(n) }[/math] points in the Euclidian plane, no three of which are collinear, has a subset of [math]\displaystyle{ n }[/math] points forming a convex [math]\displaystyle{ n }[/math]-gon.