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Point Configurations in d-Space without Large Subsets in Convex Position

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In this paper we give a lower bound for the Erd\H os–Szekeres number in higher dimensions. Namely, in two different ways we construct, for every $n>d\ge 2$, a configuration of $n$ points in general position in $\R^d$ containing at most $c_d(\log n)^{d-1}$ points in convex position. (Points in $\R^d$ are in convex position if none of them lies in the convex hull of the others.)


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Correspondence to Gyula Károlyi or Pavel Valtr.

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Károlyi, G., Valtr, P. Point Configurations in d-Space without Large Subsets in Convex Position. Discrete Comput Geom 30, 277–286 (2003).

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  • Convex Hull
  • High Dimension
  • General Position
  • Large Subset
  • Point Configuration