On the dimension of \(n\)-point sets
The paper is related to the following old result (S. Mazurkiewicz, 1914): There exists a subset of the plane which intersects every straight line in exactly two points. A subset \(S\) of the plane is an \(n\)-point set (a partial \(n\)-point set) provided that \(S\) intersects every straight line in exactly \(n\) points (at most \(n\) points). The authors answer a question asked by Bouhjar and Dijkstra, whether or not every one-dimensional partial \(n\)-point set contains an arc. The answer is affirmative; moreover, they prove the following (Th. 2.3). Let \(S\) be a subset of the plane which intersects every straight line in only a finite number of points. Then \(S\) is one-dimensional if and only if \(S\) contains an arc.
scientific article; zbMATH DE number 1901919