On multivalued fixed-point free maps on $\mathbb{R}^n$

To formulate our results let $f$ be a continuous multivalued map from $\mathbb{R}^n$ to $2^{\mathbb{R}^n}$ and $k$ a natural number such that $\vert f(x)\vert\leq k$ for all $x$. We prove that $f$ is fixed-point free if and only if its continuous extension $\tilde f:\beta \mathbb{R}^n\to 2^{\beta \mathbb{R}^n}$ is fixed-point free. If one wishes to stay within metric terms, the result can be formulated as follows: $f$ is fixed-point free if and only if there exists a continuous fixed-point free extension $\bar f: b\mathbb{R}^n\to 2^{b\mathbb{R}^n}$ for some metric compactificaton $b\mathbb{R}^n$ of $\mathbb{R}^n$. Using the classical notion of colorablity, we prove that such an $f$ is always colorable. Moreover, a number of colors sufficient to paint the graph can be expressed as a function of $n$ and $k$ only. The mentioned results also hold if the domain is replaced by any closed subspace of $\mathbb{R}^n$ without any changes in the range.