Coloring graphs from lists with bounded size of their union

A graph G is k-choosable if its vertices can be colored from any lists L(v) of colors with jL(v)j k for all v 2 V (G). A graph G is said to be (k; u)-choosable if its vertices can be colored from any lists L(v) with jL(v)j k, for all v 2 V (G), and with jSv2V (G) L(v)j u. For each 3 k u, we construct a graph G which is (k; u)-choosable but not (k; u+1)-choosable. On the other hand, it is proven that each (k; 2k 1)choosable graph G is O(k ln k 24k)-choosable.