Geodesic
On Erd\H os--Szekeres problem for empty hexagons in the plane
Modelirovanie i analiz informacionnyh sistem, Tome 16 (2009) no. 2, pp. 22-74

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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 $h(6)\le 463$.
Keywords: general position, Ramsey theory.
Mots-clés : convex polygons 
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     author = {V. A. Koshelev},
     title = {On {Erd\H} {os--Szekeres} problem for empty hexagons in the plane},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {22--74},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2009_16_2_a1/}
}
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V. A. Koshelev. On Erd\H os--Szekeres problem for empty hexagons in the plane. Modelirovanie i analiz informacionnyh sistem, Tome 16 (2009) no. 2, pp. 22-74. http://geodesic.mathdoc.fr/item/MAIS_2009_16_2_a1/