Key points are not available for this paper at this time.
It is shown that the following holds for each >0. For G an n-vertex graph of maximum degree D, lists Sᵥ (v V (G) ), and Lᵥ chosen uniformly from the ( (1+) n) -subsets of Sᵥ (independent of other choices), \ G admits a proper coloring with ᵥ Lᵥ v \ with probability tending to 1 as D. When each Sᵥ is \1 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.
Kahn et al. (Tue,) studied this question.