Journal of Automated Reasoning

, Volume 62, Issue 3, pp 301–329 | Cite as

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

  • Filip MarićEmail author


A conjecture originally made by Klein and Szekeres in 1932 (now commonly known as “Erdős–Szekeres” or “Happy Ending” conjecture) claims that for every \(m \ge 3\), every set of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.


Erdős–Szekeres conjecture Happy ending problem Convex polygons Interactive theorem proving SAT solving Isabelle/HOL 


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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