  This thesis addresses the question of the maximal number of $d$-simplices for
a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r
\leq 2d$.
  A lower bound of $f_d(C_{r + 1}(n)) = \Omega(n^{\lceil\frac{r}{2}\rceil})$,
which might even be sharp, is given by the cyclic polytopes. To find an upper
bound for the case $r=2d$ we look for forbidden subcomplexes. A generalization
of the theorem of van Kampen and Flores yields those. Then the problem can be
tackled with the methods of extremal hypergraph theory, which gives an upper
bound of $O(n^{d+1-\frac{1}{3^d}})$.
  We also consider whether these bounds can be improved by simple means.
