  Assume $ k $ is a positive integer, $ \lambda=\{k_1,k_2,...,k_q\} $ is a
partition of $ k $ and $ G $ is a graph. A $\lambda$-assignment of $ G $ is a $
k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)}
L(v) $ can be partitioned into $ q $ subsets $ C_1\cup C_2\cup\cdots\cup C_q $
and for each vertex $ v $ of $ G $, $ |L(v)\cap C_i|=k_i $. We say $ G $ is
$\lambda$-choosable if for each $\lambda$-assignment $ L $ of $ G $, $ G $ is $
L $-colourable. In particular, if $ \lambda=\{k\} $, then $\lambda$-choosable
is the same as $ k $-choosable, if $ \lambda=\{1, 1,...,1\} $, then
$\lambda$-choosable is equivalent to $ k $-colourable. For the other partitions
of $ k $ sandwiched between $ \{k\} $ and $ \{1, 1,...,1\} $ in terms of
refinements, $\lambda$-choosability reveals a complex hierarchy of
colourability of graphs.
  Assume $\lambda=\{k_1, \ldots, k_q\} $ is a partition of $ k $ and $\lambda'
$ is a partition of $ k'\ge k $. We write $ \lambda\le \lambda' $ if there is a
partition $\lambda''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for
$i=1,2,\ldots, q$ and
  $\lambda'$ is a refinement of $\lambda''$. It follows from the definition
that if $ \lambda\le \lambda' $, then every $\lambda$-choosable graph is
$\lambda'$-choosable. It was proved in [X. Zhu, A refinement of choosability of
graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is
also true. This paper strengthens this result and proves that for any $
\lambda\not\le \lambda' $, for any integer $g$, there exists a graph of girth
at least $g$ which is $\lambda$-choosable but not $\lambda'$-choosable.
