I'm studying the paper mentioned above. So we are looking at the graph $G$, which vertices are all points of $\mathbb{Q}^4$ and two vertices have an edge iff they have euclidean distance equal to $1$. Now any two points with an edge must have different colors. The chromatic number $\chi$ of a graph is the smallest number of colors needed to color the graph. The paper shows $\chi(G)=4$. The set $\mathcal{H}^4$ describes all points connected to $(0,0,0,0)$, which is a subgroup of $\mathbb{Q}^4$ under coordinate-wise addition. The index of the group $\mathcal{H}^4$ in $\mathbb{Q}^4$, noted by $[\mathbb{Q}^4 : \mathcal{H}^4]$, then measures the degree of unconnectedness of $G$, since the cosets are the unconnected components of $G$.
Here is a link to the paper: https://researchgate.net/publication/268311008
Now the paper claims, the cosets are determined by e.g. the points $(\frac{1}{2^n},0,0,0)$ with $n>1$ and therefore $[\mathbb{Q}^4 : \mathcal{H}^4] = \infty$. Can now anyone explain to me, why the cosets are determined in this way?
Remark: In the case of $\mathbb{Q}^2$ and $\mathbb{Q}^3$, the chromatic number is 2 and therefore it is bipartite, so there are no odd cycles in the graph. In this case, the author uses cycles of odd lenght to proof that $(\frac{1}{2^n},0)$ and $(\frac{1}{2^m},0)$ cannot be connected if $m \neq n$. But actually in the case of dimension four we have $\chi(G) = 4$ and therefore this argument does not work anymore.
I hope anyone can help me, although it is a detailed question.