  We present results referring to the Hadwiger-Nelson problem which asks for
the minimum number of colours needed to colour the plane with no two points at
distance $1$ having the same colour. Exoo considered a more general problem
concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two
vertices adjacent if their distance is in the interval $[a,b]$. Exoo
conjectured $\chi(G_{[a,b]}) = 7$ for sufficiently small but positive
difference between $a$ and $b$. We partially answer this conjecture by proving
that $\chi(G_{[a,b]}) \geq 5$ for $b > a$.
  A $j$-fold colouring of graph $G = (V,E)$ is an assignment of $j$-elemental
sets of colours to the vertices of $G$, in such a way that the sets assigned to
any two adjacent vertices are disjoint. The fractional chromatic number
$\chi_f(G)$ is the infimum of fractions $k/j$ for $j$-fold colouring of $G$
using $k$ colours. We generalize a method by Hochberg and O'Donnel (who proved
that $G_{[1,1]} \leq 4.36$) for fractional colouring of graphs $G_{[a,b]}$,
obtaining a bound dependant on $\frac{a}{b}$.
  We also present few specific and two general methods for $j$-fold coloring of
$G_{[a,b]}$ for small $j$, in particular for $G_{[1,1]}$ and $G_{[1,2]}$. The
$j$-fold colouring for small $j$ has strong practical motivation especially in
scheduling theory, while graph $G_{[1,2]}$ is often used to model hidden
conflicts in radio networks.

Combinatorics