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Let $G$ be a plane graph with $C$ the boundary of the outer face and let$(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of asubgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that$\phi(v)\in L(v)$ for every vertex $v$ of $J$. Thomassen proved that if$v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every$v\in V(C)\setminus \{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)\setminusV(C)$, then $G$ has an $L$-coloring.  As one final application in this last part of our series on$5$-list-coloring, we derive from all of our theory a far-reachinggeneralization of Thomassen's theorem, namely the generalization of Thomassen'stheorem to arbitrarily many such faces provided that the faces are pairwisedistance $D$ apart for some universal constant $D>0$.

Five-List-Coloring Graphs on Surfaces: The Many Faces Far-Apart Generalization of Thomassen's Theorem