Sets with No Empty Convex 7-Gons

Abstract

Erdös has defined g(n) as the smallest integer such that any set of g(n) points in the plane, no three collinear, contains the vertex set of a convex n-gon whose interior contains no point of this set. Arbitrarily large sets containing no empty convex 7-gon are constructed, showing that g(n) does not exist for n≥l. Whether g(6) exists is unknown.

On the Number of Edges in Geometric Graphs Without Empty Triangles

“…Horton sets will be key elements for achieving this result. Horton [5] recursively constructed a set H k of size 2 k , for any positive integer k such that H k has no 7-holes. The construction is as follows:…”

“…Erdős [2] asked about the existence of k-holes in planar point sets. Horton [5] proved that for k ≥ 7 there are point sets containing no k-holes. Nicolás [7] proved that any point set with sufficiently many points contains a 6-hole.…”

Almost empty monochromatic triangles in planar point sets

“…Esther Klein showed that H(4) = 5 and Harborth [14] proved that H(5) = 10. Horton [15] showed that it is possible to construct an arbitrarily large set of points without a 7-hole, thereby proving that H(k) does not exist for k ≥ 7. After a long wait, the existence of H(6) was proved by Gerken [12] and independently by Nicolás [22].…”

Внутренние точки в теоремах Эрдeша - Секереша

“…[13]). Именно в связи с этим возникает третья проблема и, в частности, вопрос о существовании величины ℎ( , ) при > 7.…”

“…В этой статье с использованием множества Хортона (см. [13]), с помощью которого было доказано несуществование ℎ(7), доказывается несущество-вание ℎ( , ) для определенных при > 7. Аналогичные результаты получены в статье Ныкловой [15]; кроме того, там доказано, что ℎ(6, 6) = (6), и представлен результат ℎ(6, 5) = 19.…”