A003186 -id:A003186 - 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.

3, 5, 10, 30

Offset: 3

Paolo Xausa, Mar 07 2025

An empty n-gon does not contain any points (also called n-hole).
This problem was posed by Erdös and Szekeres (1935), generalizing on a result by Esther Klein.
For n >= 7, such n-gons can be avoided (this result is due to Horton, 1983).
The case for n = 6 was solved by Heule and Scheucher (2024).

P. Erdös and G. Szekeres, A Combinatorial Problem in Geometry, Compositio Mathematica, Volume 2 (1935), pp. 463-470 (alternative source).
Marijn J. H. Heule and Manfred Scheucher, Happy Ending: An Empty Hexagon in Every Set of 30 Points, arXiv:2403.00737 [cs.CG], 2024.
J. D. Horton, Sets with No Empty Convex 7-Gons, Canadian Mathematical Bulletin, Vol. 26 Issue 4, 1983, pp. 482-484.
PurpleMind, The Miracle Solution to This 100 Year Old Geometry Problem, YouTube video, 2025.
Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro and Marijn J. H. Heule, Formal Verification of the Empty Hexagon Number, arXiv:2403.17370 [cs.CG], 2024.
Wikipedia, Happy ending problem.

Cf. A003182, A003186, A330333, A348260.

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.

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