A381776 - OEIS

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

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OFFSET

3,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=3..6.

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.

CROSSREFS

Cf. A003182, A003186, A330333, A348260.

Sequence in context: A262306 A284301 A284349 * A003186 A006826 A000214

Adjacent sequences: A381773 A381774 A381775 * A381777 A381778 A381779

KEYWORD

nonn,bref,fini,full,nice

AUTHOR

Paolo Xausa, Mar 07 2025

STATUS

approved