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Updated Oct 8

Problem 216

claims/: The 5 claim pages of Problem 216, one per claimant's result; the problem's standing derives from them.

Statement. Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $\mathbb{R}^2$ contains an empty convex $k$ -gon (i.e. with no point in the interior). Does $g(k)$ exist? If so, estimate $g(k)$ .

Formulation. The site's wording omits the convention, standard for $g(k)$ , that the points are in general position, no three on a line. Read as the site words it, $g(k)$ exists for no $k\ge3$ , since any number of collinear points contains no convex $k$ -gon. The page reads the problem as Erdős's 1978 formulation (Austral. Math. Soc. Gaz. 5 (1978), p. 53) and the sources do, over sets with no three points collinear (Horton's definition of $g(n)$ , Harborth, Nicolás, Gerken, Heule and Scheucher), which is the reading of every claim page. The negative answer is the same under both readings; the values $g(4)=5$ , $g(5)=10$ and $g(6)=30$ hold only under this one.

Status. Disproved: the site labels the problem disproved on Horton's construction of arbitrarily large point sets with no empty convex heptagon, recorded on the Horton claim page. The site's remarks credit the instances that exist: $g(4)=5$ (Erdős, no source given, so it stays in prose), $g(5)=10$ on the Harborth claim page, the existence of $g(6)$ on the Nicolás and Gerken claim pages, and $g(6)=30$ on the Heule–Scheucher claim page.

Source. erdosproblems.com/216, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #216, https://www.erdosproblems.com/216.

References.

[Ge08] Gerken, Tobias, Empty convex hexagons in planar point sets. Discrete Comput. Geom. (2008), 239-272.
[Ha78] Harborth, Heiko, Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. (1978), 116-118.
[HeSc24] M. Heule and M. Scheucher, Happy Ending: An Empty Hexagon in Every Set of $30$ Points. Tools and Algorithms for the Construction and Analysis of Systems, Lecture Notes in Computer Science (2024), 61-80.
[Ho83] Horton, J. D., Sets with no empty convex 7-gons. Canad. Math. Bull. (1983), 482-484.
[Ni07] Nicolás, Carlos M., The empty hexagon theorem. Discrete Comput. Geom. (2007), 389-397.

Formalization. None recorded in the community database. A Lean verification of Heule and Scheucher's $g(6)=30$ by Subercaseaux, Nawrocki, Gallicchio, Codel, Carneiro and Heule is linked from their claim page; the corpus has not built it.

Progress

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Linked library material

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Linked from (13)

Harborth's empty convex pentagon in every 10 points Nicolás's empty hexagon theorem Gerken's empty convex hexagon from a convex 9-gon problems/discrete_geometry/E0216/claims Discrete and Convex Geometry Discrete and Convex Geometry discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number The empty convex polygon question, p. 138: n_4 = 5, n_5 = 10, and whether n_6 exists discrete_geometry/heule_2024_happy_ending_empty_hexagon_30_points discrete_geometry/horton_1983_sets_no_empty_convex_7_gons Main theorem (p. 482): the Horton set S_k of 2^k points has no empty convex 7-gon, so g(7) does not exist distance_problems/erdos_1983_combinatorial_problems_geometry Empty convex polygons (pp. 51--52): n(k), with n(4) = 5, Harborth's n(5) = 10, and the open existence of n(6)

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