Tuesday, November 22, 16.30. Mathematical Institute, University of Bonn, room 0.003

Wolfgang Wohofsky (Vienna): Small subsets of the real line and variants of the Borel Conjecture

Abstract: I will first give a short overview of (the history of) the notions “strong measure zero” and “strongly meager”, and of the consistency proofs of the Borel Conjecture (by Laver) and the dual Borel Conjecture (by Carlson).

Then I will talk about our recent result that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold

simultaneously. This is joint work with Martin Goldstern, Jakob Kellner, and Saharon Shelah. I will try to present at least some of the ideas involved in the proof.

Finally, I will talk about another variant of the Borel Conjecture, which I call “Marczewski Borel Conjecture” (MBC). It is the analogue of the (dual) Borel Conjecture when the ideal of meager (measure zero) sets in its definition is replaced by the ideal of Marczewski null sets. I still do not know whether it is consistent; to investigate this question, I introduced the notion of “Sacks dense ideal”: I will discuss its relation to MBC and outline several results (and open problems) about Sacks dense ideals.