On subsets of the plane that intersect every line in a fixed number of points
by
J. van Mill
Vrije Universiteit

Let n >= 2. A subset X of the plane is called an n-point set if it intersects every line in exactly n-points. (This concept does not make sense if n=1 and an increasing union of concentric circles shows it is not interesting for infinite n). Mazurkiewicz proved that n-point sets exist for every n. Whether such a set can be Borel is a formidable open problem. The only known result is that it cannot be an F_\sigma-subset of the plane by a recent theorem of Bouhjar, Dijkstra and Mauldin. Kulesza proved that a 2-point set is zero-dimensional. It is unknown whether a 3-point set must be zero-dimensional. We prove that there are 4-point sets containing arcs. We also discuss so-called weak 2-point sets and prove that they are zero-dimensional, are G_\delta if they are F_\sigma, and that they cannot be dense in the plane.

Date received: June 5, 2000