Partial list colorings

Suppose G is an s-choosable graph with n vertices, and every vertex of G is assigned a list of t colors. We conjecture that at least (t/s)n of the vertices of G can be colored from these lists. We provide lower bounds and consider related questions. For instance, we show that if G is χ-colorable (rather than being s-choosable), then more than (1−((χ−1)/χ)t)n of the vertices of G can be colored from the lists and that this is asymptotically best possible. We include a number of open questions.