$n$ points on every line For which integers $n$ is it possible to find a subset $S$ of $\mathbb R^2$ such that every infinite line contains exactly $n$ points of $S$?

According to Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 10, problem 21, page 144, "F. Galvin has shown the following: if $1\lt n\le\aleph_0$ then there is a non-measurable set in the plane such that the intersection of $S$ with any line consists of precisely $n$ points." Unfortunately, no bibliographic details are given. 
