"Paradoxical" sets with no well-ordering of the reals

By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any model in which R cannot be well ordered." About a year ago, we answered this positively in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint work, additionally with J. Brendle and F. Castiblanco, we constructed a model of ZF plus DC with a Luzin set, a Sierpiński set, a Burstin basis, and a Mazurkiewicz set, but with no well-ordering of R. We will give an outline of the constructions. A superset of the content of the talk is available at: https://ivv5hpp.uni-muenster.de/u/rds/everything.pdf and https://ivv5hpp.uni-muenster.de/u/rds/groszek_slaman.pdf