3 problems

Let $d$ and $q$ be positive integers, let $\Delta^{(d+1)(q-1)}_d$ be the $d$-skeleton of the $((d+1)(q-1))$-dimensional simplex, and let $f:\Delta^{(d+1)(q-1)}_d\to\mathbb{R}^d$ be…

Let $d\geq 1$ and $r\geq 2$ be integers, let $N=(d+1)(r-1)$, and let $f:\Delta_N\to\mathbb{R}^d$ be continuous. A collection of faces is pairwise disjoint if the faces have no comm…

Blagojević–Frick–Ziegler's conjecture. There are $r$ pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $\Delta_N$ such that