  A graph G is (a:b)-colorable if there exists an assignment of b-element
subsets of {1,...,a} to vertices of G such that sets assigned to adjacent
vertices are disjoint. We first show that for every triangle-free planar graph
G and a vertex x of G, the graph G has a set coloring phi by subsets of
{1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a
corollary, every triangle-free planar graph on n vertices is
(6n:2n+1)-colorable. We further use this result to prove that for every Delta,
there exists a constant M_Delta such that every planar graph G of girth at
least five and maximum degree Delta is (6M_Delta:2M_Delta+1)-colorable.
Consequently, planar graphs of girth at least five with bounded maximum degree
Delta have fractional chromatic number at most 3-3/(2M_Delta+1).
