  The theory of finitely supported algebraic structures represents a
reformulation of Zermelo-Fraenkel set theory in which every construction is
finitely supported according to the action of a group of permutations of some
basic elements named atoms. In this paper we study the properties of finitely
supported sets that contain infinite uniformly supported subsets, as well as
the properties of finitely supported sets that do not contain infinite
uniformly supported subsets. For classical atomic sets, we study whether they
contain or not infinite uniformly supported subsets.
