Point Configurations in d-Space without Large Subsetsin Convex Position

In this paper we give a lower bound forthe ErdH os–Szekeres number in higher dimensions.Namely, in two different wayswe construct, for every $n>dge 2$,a configuration of $n$ pointsin general position in $R^d$ containingat most $c_d(log n)^{d-1}$ points in convex position.(Points in $R^d$ are in convex position ifnone of them lies in the convex hull of the others.)