The authors state and prove the following result.: Let \(G\) be a planar graph with no cycles of length \(4\) or \(6\). Let \(k\geq 6\) be an integer, which is also greater or equal to the maximum degree of \(G\). Then for every ``\(k\)--uniform list assignment'' (this is an assignment of finite sets of \(k\) admissible colours to each vertex of \(G\)), there is a proper vertex colouring of \(G\) such that no block of the partition of \(V(G)\) induced by that colouring contains more than \(\lceil{V(G)\over k}\rceil\) vertices.