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Fast Formal Proof of the Erdős–Szekeres Conjecture for Convex Polygons with at Most 6 Points

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Abstract

A conjecture originally made by Klein and Szekeres in 1932 (now commonly known as “Erdős–Szekeres” or “Happy Ending” conjecture) claims that for every \(m \ge 3\), every set of \(2^{m-2}+1\) points in a general position (none three different points are collinear) contains a convex m-gon. The conjecture has been verified for \(m \le 6\). The case \(m=6\) was solved by Szekeres and Peters and required a huge computer enumeration that took “more than 3000 GHz hours”. In this paper we improve the solution in several directions. By changing the problem representation, by employing symmetry-breaking and by using modern SAT solvers, we reduce the proving time to around only a half of an hour on an ordinary PC computer (i.e., our proof requires only around 1 GHz hour). Also, we formalize the proof within the Isabelle/HOL proof assistant, making it significantly more reliable.

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Notes

  1. The name was coined by Paul Erdős, since the original problem posed for \(m=4\) and \(n=5\) by Esther Klein led to her marriage to George Szekeres.

  2. This is the standard input format for SAT solvers.

  3. The patch was taken from the SAT 2014 competition website http://www.satcompetition.org/2014/description.shtml.

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  1. Faculty of Mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Serbia

    Filip Marić

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  1. Filip Marić
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Correspondence to Filip Marić.

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This work has been partially supported by the Grant 174021 of the Ministry of Science of Serbia.

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Marić, F. Fast Formal Proof of the Erdős–Szekeres Conjecture for Convex Polygons with at Most 6 Points. J Autom Reasoning 62, 301–329 (2019). https://doi.org/10.1007/s10817-017-9423-7

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