On the Complexity of Embeddable Simplicial Complexes

Understand

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.