Improper choosability and Property B

A fundamental connection between list vertex colourings of graphs and Property B (also known as hypergraph 2-colourability) was already known to Erd\H{o}s, Rubin and Taylor. In this article, we draw similar connections for improper list colourings. This extends results of Kostochka, Alon, and Kr\'al' and Sgall for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.