Two Geometrical Applications of the Semi-random Method

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 $\Omega \left( \frac{n^2\log n}{\log \log n}\right)$ 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 $\mathcal {O}(n^{-3/2}(\log n)^{1/2})$.

Partially supported by TÉT_12_MX-1-2013-0006 and by National Research, Development and Innovation Office – NKFIH Fund No. SNN-117879. Supported by ERC-AdG. 321104, and OTKA Grant NK104186.