turkmath.org

In the first part of this talk, we shall first discuss some classical constructions of “implicit” nature, such as a paradoxical decomposition of the unit ball, a Hamel basis and a well-ordering of R, maximal independent families of N etc. We shall then focus on the following question: Can these constructions be carried out in an “explicit” way? In the second part of this talk, we shall focus on two-point sets. A subset of the plane is called a two-point set if it intersects every line at exactly two points. It is a well-known result of Mazurkiewicz that two-point sets exist. On the other hand, whether or not such a set can be Borel has been open for a long time. We shall discuss this problem and what has been done so far.