On the Number of Simplicial Complexes in D

Abstract Using a simplex-crossing counting technique we prove: if the number of non-improperly intersecting simplices with vertices in a set S of n labeled points in Rd is O (n ⌈ d 2 ⌉ ) , then there are 2 Θ(n ⌈ d 2 ⌉ ) different geometric simplicial complexes with vertices in S.