On the Complexity of Embeddable Simplicial Complexes

TL;DR

This thesis investigates the maximum number of simplices in embeddable complexes within certain dimensions, providing bounds using extremal hypergraph theory and exploring potential improvements.

Contribution

It introduces new bounds on the number of simplices in embeddable complexes and generalizes classical theorems to higher dimensions.

Findings

Lower bound of _d(C_{r + 1}(n)) = \u03a9(n^{eilinga9r/2})

Upper bound of O(n^{d+1-/3^d}) using extremal hypergraph theory

Potential for improving these bounds through simple methods

Abstract

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.