# New Trends in Intuitive GeometryTwo Geometrical Applications of the Semi-random Method

New Trends in Intuitive Geometry: Two Geometrical Applications of the Semi-random Method [The semi-random method was introduced in the early eighties. In its first form of the method lower bounds were given for the size of the largest independent set in hypergraphs with certain uncrowdedness properties. The first geometrical application was a major achievement in the history of Heilbronn’s triangle problem. It proved that the original conjecture of Heilbronn was false. The semi-random method was extended and applied to other problems. In this paper we give two further geometrical applications of it. First, we give a slight improvement on Payne and Wood’s upper bounds on a Ramsey-type parameter, introduced by Gowers. We prove that any planar point set of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \left( \frac{n^2\log n}{\log \log n}\right)$$\end{document} contains n points on a line or n independent points. Second, we give a slight improvement on Schmidt’s bound on Heilbronn’s quadrangle problem. We prove that there exists a point set of size n in the unit square that doesn’t contain four points with convex hull of area \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(n^{-3/2}(\log n)^{1/2})$$\end{document}.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# New Trends in Intuitive GeometryTwo Geometrical Applications of the Semi-random Method

Part of the Bolyai Society Mathematical Studies Book Series (volume 27)
Editors: Ambrus, Gergely; Bárány, Imre; Böröczky, Károly J.; Fejes Tóth, Gábor ; Pach, János
10 pages

/lp/springer-journals/new-trends-in-intuitive-geometry-two-geometrical-applications-of-the-06YS0eGoxy
Publisher
Springer Berlin Heidelberg
© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018. Corrected Publication 2018
ISBN
978-3-662-57412-6
Pages
189 –199
DOI
10.1007/978-3-662-57413-3_8
Publisher site
See Chapter on Publisher Site

### Abstract

[The semi-random method was introduced in the early eighties. In its first form of the method lower bounds were given for the size of the largest independent set in hypergraphs with certain uncrowdedness properties. The first geometrical application was a major achievement in the history of Heilbronn’s triangle problem. It proved that the original conjecture of Heilbronn was false. The semi-random method was extended and applied to other problems. In this paper we give two further geometrical applications of it. First, we give a slight improvement on Payne and Wood’s upper bounds on a Ramsey-type parameter, introduced by Gowers. We prove that any planar point set of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \left( \frac{n^2\log n}{\log \log n}\right)$$\end{document} contains n points on a line or n independent points. Second, we give a slight improvement on Schmidt’s bound on Heilbronn’s quadrangle problem. We prove that there exists a point set of size n in the unit square that doesn’t contain four points with convex hull of area \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(n^{-3/2}(\log n)^{1/2})$$\end{document}.]

Published: Nov 3, 2018