'Two-point sets and the Axiom of Choice'.

'A two-point set is a subset of the plane which meets every line in exactly two points. The existence of two-point sets was shown by Mazurkiewicz in 1914, and the main open problem concerning these objects is to determine if there exist Borel two-point sets. If this question has a positive answer, then we most likely need to be able to construct a two-point set without making use of a well-ordering of the real line, as is currently the usual technique.

We discuss recent work by Robin Knight, Rolf Suabedissen and the speaker, and (independently) by Arnold Miller, which show that it is consistent with ZF that the real line cannot be well-ordered and also that two-point sets exist.'