This theorem is originally to S. Mazurkiewicz. In Sierpinski's book "Cardinal and Ordinal Numbers", the theorem is proved on page 449 (Chapter 17, Section 2). In fact the following more general result by F. Bagemihl ("A theorem on intersections of prescribed cardinality", Annals of Math., 55 (1952), p. 34, jstor) holds:
Theorem: Suppose that every straight line $S$ lying in a plane is associated with cardinal number $m_S$ such that $2 \leq m \leq 2^{\aleph_0}$. Then there exists a plane set $Q$ such that the cardinality of $Q \cap S$ is $m_S$ for every straight line $S$ in the plane.
(I am mentioning Sierpinski's book because it might be more easily available than 1952 issues of Annals of Mathematics.)
