On Erdős–Szekeres problem for empty hexagons in the plane

In this work we consider a classical problem of Combinatorial Geometry of P. Erdős and G. Szekeres. The problem was posed in the 1930's. We investigate the minimum number $h(n)$ such, that for each $h(n)$-point set $A$ in general position in the plane there exists an $n$-point subset $B$ such, that the convex hull $C$ of $B$ is a convex empty $n$-gon, that is $(A\setminus B)\cap C=\emptyset$. Only recently T. Gerken has shown that $h(6)<\infty$. He has established the inequality $h(6)\le 1717$. The main result of the paper is the following inequality $h(6)\le 463$.