  In 1998, Reed conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil
\frac{1}{2}(\Delta(G) + 1 + \omega(G))\rceil$, where $\chi(G)$ is the chromatic
number of $G$, $\Delta(G)$ is the maximum degree of $G$, and $\omega(G)$ is the
clique number of $G$. As evidence for his conjecture, he proved an "epsilon
version" of it, i.e. that there exists some $\varepsilon > 0$ such that
$\chi(G) \leq (1 - \varepsilon)(\Delta(G) + 1) + \varepsilon\omega(G)$. It is
natural to ask if Reed's conjecture or an epsilon version of it is true for the
list-chromatic number. In this paper we consider a "local version" of the
list-coloring version of Reed's conjecture. Namely, we conjecture that if $G$
is a graph with list-assignment $L$ such that for each vertex $v$ of $G$,
$|L(v)| \geq \lceil \frac{1}{2}(d(v) + 1 + \omega(v))\rceil$, where $d(v)$ is
the degree of $v$ and $\omega(v)$ is the size of the largest clique containing
$v$, then $G$ is $L$-colorable. Our main result is that an "epsilon version" of
this conjecture is true, under some mild assumptions.
  Using this result, we also prove a significantly improved lower bound on the
density of $k$-critical graphs with clique number less than $k/2$, as follows.
For every $\alpha > 0$, if $\varepsilon \leq \frac{\alpha^2}{1350}$, then if
$G$ is an $L$-critical graph for some $k$-list-assignment $L$ such that
$\omega(G) < (\frac{1}{2} - \alpha)k$ and $k$ is sufficiently large, then $G$
has average degree at least $(1 + \varepsilon)k$. This implies that for every
$\alpha > 0$, there exists $\varepsilon > 0$ such that if $G$ is a graph with
$\omega(G)\leq (\frac{1}{2} - \alpha)\mathrm{mad}(G)$, where $\mathrm{mad}(G)$
is the maximum average degree of $G$, then $\chi_\ell(G) \leq \left\lceil (1 -
\varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon \omega(G)\right\rceil$.
