cp's OEIS Frontend

A381776 Empty polygon numbers: a(n) is the smallest number of points in the plane (with no three of them collinear) such that an empty convex n-gon cannot be avoided.

%I A381776 #22 Mar 07 2025 08:22:56
%S A381776 3,5,10,30
%N A381776 Empty polygon numbers: a(n) is the smallest number of points in the plane (with no three of them collinear) such that an empty convex n-gon cannot be avoided.
%C A381776 An empty n-gon does not contain any points (also called n-hole).
%C A381776 This problem was posed by Erdös and Szekeres (1935), generalizing on a result by Esther Klein.
%C A381776 For n >= 7, such n-gons can be avoided (this result is due to Horton, 1983).
%C A381776 The case for n = 6 was solved by Heule and Scheucher (2024).
%H A381776 P. Erdös and G. Szekeres, <a href="http://www.numdam.org/item/?id=CM_1935__2__463_0">A Combinatorial Problem in Geometry</a>, Compositio Mathematica, Volume 2 (1935), pp. 463-470 (<a href="https://www.renyi.hu/~p_erdos/1935-01.pdf">alternative source</a>).
%H A381776 Marijn J. H. Heule and Manfred Scheucher, <a href="https://doi.org/10.48550/arXiv.2403.00737">Happy Ending: An Empty Hexagon in Every Set of 30 Points</a>, arXiv:2403.00737 [cs.CG], 2024.
%H A381776 J. D. Horton, <a href="https://doi.org/10.4153/CMB-1983-077-8">Sets with No Empty Convex 7-Gons</a>, Canadian Mathematical Bulletin, Vol. 26 Issue 4, 1983, pp. 482-484.
%H A381776 PurpleMind, <a href="https://www.youtube.com/watch?v=0_fdjA2R0bQ">The Miracle Solution to This 100 Year Old Geometry Problem</a>, YouTube video, 2025.
%H A381776 Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro and Marijn J. H. Heule, <a href="https://doi.org/10.48550/arXiv.2403.17370">Formal Verification of the Empty Hexagon Number</a>, arXiv:2403.17370 [cs.CG], 2024.
%H A381776 Wikipedia, <a href="https://en.wikipedia.org/wiki/Happy_ending_problem">Happy ending problem</a>.
%Y A381776 Cf. A003182, A003186, A330333, A348260.
%K A381776 nonn,bref,fini,full,nice
%O A381776 3,1
%A A381776 _Paolo Xausa_, Mar 07 2025