APPLICATIONS OF THE AXIOM OF CHOICE TO CONSTRUCTIONS IN EUCLIDEAN SPACES.

We give a short account of corollaries of the Axiom of Choice, which show the existence of curious configurations or colorings of the Euclidean spaces. The paper is readable even with a minimal knowledge of set theory. A possible new result is also proved: pairwise disjoint flags can be elected on each point of the plane. [ABSTRACT FROM AUTHOR]