  It is shown that the following holds for each $\varepsilon >0$. For $G$ an
$n$-vertex graph of maximum degree $D$, lists $S_v$ ($v\in V(G)$), and $L_v$
chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$
(independent of other choices), \[ \mbox{$G$ admits a proper coloring $\sigma$
with $\sigma_v\in L_v$ $\forall v$} \] with probability tending to 1 as $D\to
\infty$.
  When each $S_v $ is $\{1\dots D+1\}$, this is an asymptotically optimal
version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna
that was proved in an earlier paper by the present authors.
