  A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not
necessarily $(c:d)$-choosable. The simplest case of another problem, stated by
the same authors, is settled, proving that every 2-choosable graph is also
$(4:2)$-choosable. Applying probabilistic methods, an upper bound for the
$k^{th}$ choice number of a graph is given. We also prove that a directed graph
with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable
for every $k \geq 1$. Other results presented in this article are related to
the strong choice number of graphs (a generalization of the strong chromatic
number). We conclude with complexity analysis of some decision problems related
to graph choosability.
