Two upper bounds for the Erdős-Szekeres number with conditions
The celebrated Erdős-Szekeres theorem asserts that there is a smallest positive integer \(ES(n)\), such that among any \(ES(n)\) points in general position in the plane, one finds the vertices of a convex \(n\)-gon, and the Erdős-Szekeres conjecture is that \(ES(n)=2^{n-2}+1\). Erdős and Szekeres proved \(ES(n)\leq \binom{2n-4}{n-2}+1\). The current best upper bound, \(ES(n)\leq \binom{2n-5}{n-2}+1\), is due to \textit{G. Tóth} and \textit{P. Valtr} [Mathematical Sciences Research Institute Publications 52, 557--568 (2005; Zbl 1090.52014)]. The paper under review uses the projective transformation method of Tóth and Valtr to prove the following results: For every \(n\geq 4\), every set of \(\binom{2n-5}{n-2}-2n+9\) points in general position in the plane with the property that there is no point inside a triangle determined by three consecutive vertices of its convex hull contains a convex \(n\)-gon. For every \(n\geq 4\), every set of \(\binom{2n-5}{n-2}-2n+10\) points in general position in the plane, whose convex hull is an \(n-1\)-gon, contains a convex \(n\)-gon.
scientific article; zbMATH DE number 6147354