  A graph G is list (b:a)-colorable if for every assignment of lists of size b
to vertices of G, there exists a choice of an a-element subset of the list at
each vertex such that the subsets chosen at adjacent vertices are disjoint. We
prove that for every positive integer a, the family of minimal obstructions of
girth at least five to list (3a:a)-colorability is strongly hyperbolic, in the
sense of the hyperbolicity theory developed by Postle and Thomas. This has a
number of consequences, e.g., that if a graph of girth at least five and Euler
genus g is not list (3a:a)-colorable, then G contains a subgraph with O(g)
vertices which is not list (3a:a) colorable.
