In this nicely written paper, the authors prove the following vast generalization of a theorem of Mazurkiewicz (1914): In any vector space \(V\) of arbitrary dimension over an infinite field \(F\) there exists a subset \(X \subseteq V\) which intersects each line in exactly \(2\) points. (Mazurkiewicz had only considered the real plane.) The assumption \(\text{card } F \geq \aleph_0\) is essential: If \(F\) is finite, \(\dim V > 1\), and if \(X\) meets each line in the same number \(n>0\) of points, then \(X=V\) (by an easy counting argument). As is to be expected, the proof of the theorem relies on the axiom of choice. It is different from the one given by Mazurkiewicz, which does not seem to work for spaces of dimension greater than \(\text{card} F\). The authors mention further results which can be proved in a similar way as the theorem above: (1) \(V\) can even be partitioned into subsets each of which has the same property as \(X\). (2) For each finite number \(n \geq 2\) there exists a subset \(X_n \subseteq V\) such that \(|X_n \cap L|= n\) for each line \(L\) of \(V\).