论文信息 - A Note on the Number of General 4-holes in (Perturbed) Grids

A Note on the Number of General 4-holes in (Perturbed) Grids

Considering a variation of the classical Erdős-Szekeres type problems, we count the number of general 4-holes (not necessarily convex, empty 4-gons) in squared Horton sets of size \(\sqrt{n}\!\times \!\sqrt{n}\). Improving on previous upper and lower bounds we show that this number is \(\varTheta (n^2\log n)\), which constitutes the currently best upper bound on minimizing the number of general 4-holes for any set of n points in the plane.

Pavel Valtr | Oswin Aichholzer | Thomas Hackl | Birgit Vogtenhuber

[1] Imre Bárány,et al. Problems and Results around the Erdös-Szekeres Convex Polygon Theorem , 2000, JCDCG.

[2] G. Szekeres,et al. A combinatorial problem in geometry , 2009 .

[3] Pavel Valtr,et al. Planar point sets with a small number of empty convex polygons , 2004 .

[4] Ruy Fabila Monroy,et al. 4-Holes in point sets , 2014, Comput. Geom..

[5] Pavel Valtr. Convex independent sets and 7-holes in restricted planar point sets , 1992, Discret. Comput. Geom..

[6] Jean-François Marckert,et al. Many Empty Triangles have a Common Edge , 2013, Discret. Comput. Geom..

[7] Dieter Mitsche,et al. Empty non-convex and convex four-gons in random point sets , 2015 .

[8] J. Horton. Sets with No Empty Convex 7-Gons , 1983, Canadian Mathematical Bulletin.

[9] Jorge Urrutia,et al. On k-Gons and k-Holes in Point Sets , 2014, CCCG.

[10] I. Bárány,et al. Empty Simplices in Euclidean Space , 1987, Canadian Mathematical Bulletin.

[11] E. Wright,et al. An Introduction to the Theory of Numbers , 1939 .

[12] P. Valtr. On the minimum number of empty polygons in planar point sets , 1992 .