Coloring face hypergraphs on surfaces

We prove that the face hypergraph of the triangulation of a surface of Euler genus g is O(g3)-choosable. This bound matches a previously known lower bound of order Ω (g3). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O(gr)-choosable. Note that colorings of face hypergraphs for r=2 correspond to usual vertex colorings and the upper bound O(g) thus follows from Heawood's formula. Separate results for small genera are presented: the bound 3 for triangulations of the surface of Euler genus g=3 and the bound 7+36g+496 for surfaces of Euler genus g≥3. Our results dominate the previously known bounds for all genera except for g=4,7,8,9,14.