List colourings of planar graphs

A graph G = G( V, E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex u is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(u) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erd&, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable. We will prove the second conjecture.