In an interesting paper [2] Ginsburg and Linek considered the question whether the plane can be covered with 3 stars where a star consists of a point and a segment of ÿnite length on every half-line from that point. We have not been able to prove this conjecture, but under the continuum hypothesis we show the stronger statement that the plane is the union of three clouds, where a cloud consists of a point plus ÿnitely many points on every half-line emanating from that point. In fact, the continuum hypothesis is equivalent with the existence of such a covering. We also show that if the continuum is at most ℵ n (for some positive natural number n) then the plane is the union of n + 2 clouds, but we are unable to show equivalence in this general case. Yet another variant is the notion of a circle, that is, a point plus one point on every half-line from that point. We show, in ZFC, that the plane is the union of countably many circles. (A simple deduction of this, from results on almost-disjoint sets, will also be given.) We conjecture that this is not possible with ÿnitely many circles, but we can only prove this if the centers of the circles are distinct.
Deÿnition. If a is a point on the plane, then a cloud around a is a set A which intersects every line e with a ∈ e in a ÿnite set.
Theorem 1. If the continuum hypothesis holds then the plane is the union of three clouds.
Proof. Assume that a, b, c are noncollinear points. We deÿne the clouds A, B, C around a, b, c by transÿnite recursion. Let E be the set of all lines going through one of a, b, or c. By closure we can ÿnd the increasing, continuous, countable decompositions