On kk-Gons and kk-Holes in Point Sets

We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position

Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s ≥ 1, let M3 (c, s) be the least integer such that any set of at least M3 (c, s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0 , which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3 (1, 0) = 3, M3 (2, 0) = 9, and M3 (c, 0) = ∞, for c ≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3 (c) such that M3 (c, λ3 (c)) \u3c ∞ satisfies: ⌊(c-1)/2⌋ ≤ λ3 (c) ≤ c - 2, where c ≥ 2. Moreover, the exact values of M3 (c, s) are determined for small values of c and s. We also conjecture that λ3 (4) = 1, and verify it for sufficiently large Horton sets

A Superlinear Lower Bound on the Number of 5-Holes

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted

On weighted sums of numbers of convex polygons in point sets

The version of record of this article, first published in Discrete & Computational Geometry, is available online at Publisher’s website: https://doi.org/10.1007/s00454-022-00395-8Let S be a set of n points in general position in the plane, and let Xk,(S) be the number of convex k-gons with vertices in S that have exactly points of S in their interior. We prove several equalities for the numbers Xk,(S). This problem is related to the Erd¿os–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.Research of C.H. was partially supported by project MTM2015-63791-R (MINECO/ FEDER), PID-2019-104129GB-I00/MCIN/AEI/10.13039/501100011033, and by project Gen. Cat. DGR 2017SGR1336. D.O. was partially supported by project PAPIIT IG100721 and CONACyT 282280. P. P-L. was partially supported by project DICYT 041933PL Vicerrectoría de Investigación, Desarrollo e Innovación USACH (Chile), and Programa Regional STICAMSUD 19-STIC-02. Research of B.V. was partially supported by the Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF Project I 3340-N35. We thank an anonymous referee for helpful comments. This project has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922Postprint (author's final draft