The continuum hypothesis is equivalent to the assertion that the points of the plane can be 2-colored such that each horizontal line hits at most countably many red points and each vertical line hits at most countably many blue points. We will discuss various generalizations of this theorem. We will prove a characterization of the cardinality of a set using orthogonal hyperplanes. We will also show an equivalence between the size of the continuum and the existence of certain partitions of lines and points in Euclidean space.