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Computer solution to the 17-point Erdős-Szekeres problem

Published online by Cambridge University Press:  17 February 2009

George Szekeres  and
Lindsay Peters
George Szekeres
Affiliation:
(deceased), School of Mathematics, University of NSW, Sydney NSW 2052, Australia.
Lindsay Peters
Affiliation:
Pacific Knowledge Systems, Australian Technology Park, Eveleigh NSW 1430, Australia; e-mail: l.peters@pks.com.au.
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Abstract

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We describe a computer proof of the 17-point version of a conjecture originally made by Klein-Szekeres in 1932 (now commonly known as the “Happy End Problem”) that a planar configuration of 17 points, no 3 points collinear, always contains a convex 6-subset. The proof makes use of a combinatorial model of planar configurations, expressed in terms of signature functions satisfying certain simple necessary conditions. The proof is more general than the original conjecture as the signature functions examined represent a larger set of configurations than those which are realisable. Three independent implementations of the computer proof have been developed, establishing that the result is readily reproducible.

Keywords

Erdos-Szekeres problemRamsey theoryconvex polygons and polyhedrageneralized convexity
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 2 , October 2006 , pp. 151 - 164
Copyright
Copyright © Australian Mathematical Society 2006

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