  In this paper we shall give a short proof of the result originally obtained
by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$
full in $A$ such that no distance between two distinct points from $B$ is
rational. We will construct a Bernstein subset of $\mathbb{R}$ which also
avoids rational distances. We will show some cases in which the former result
may be extended to subsets of $\mathbb{R}^2$, i. e. it remains true for
measurable subsets of the plane and if $non(\mathcal{N})=cof(\mathcal{N})$ then
for a given set of positive outer measure we may find its full subset which is
a partial bijection and avoids rational distances.
