  Thomassen proved that every planar graph $G$ on $n$ vertices has at least
$2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at
least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for
$G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a
graph on $n$ vertices embedded on a surface $\Sigma$ of genus $g$, then there
exist constants $\epsilon,c_g > 0$ such that if $G$ has an $L$-coloring, then
$G$ has at least $c_g2^{\epsilon n}$ distinct $L$-colorings if $L$ is a
5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has
girth at least five. More generally, they proved that there exist constants
$\epsilon,\alpha>0$ such that if $G$ is a graph on $n$ vertices embedded in a
surface $\Sigma$ of fixed genus $g$, $H$ is a proper subgraph of $G$, and
$\phi$ is an $L$-coloring of $H$ that extends to an $L$-coloring of $G$, then
$\phi$ extends to at least $2^{\epsilon(n - \alpha(g + |V(H)|))}$ distinct
$L$-colorings of $G$ if $L$ is a 5-list-assignment or if $L$ is a
3-list-assignment and $G$ has girth at least five. We prove the same result if
$G$ is triangle-free and $L$ is a 4-list-assignment of $G$, where
$\epsilon=\frac{1}{8}$, and $\alpha= 130$.
