An (a:b)-coloring of a graph G is a function f which maps the vertices
of G into b-element subsets of some set of size a in such a way that
f(u) is disjoint from f(v) for every two adjacent vertices u and v in
G. The fractional chromatic number Οfβ(G) is the infimum of a/b over
all pairs of positive integers a,b such that G has an (a:b)-coloring.
Heckman and Thomas conjectured that the fractional chromatic number of every
triangle-free graph G of maximum degree at most three is at most 2.8. Hatami
and Zhu proved that Οfβ(G)β€3β3/64β2.953. Lu and Peng improved
the bound to Οfβ(G)β€3β3/43β2.930. Recently, Ferguson, Kaiser
and Kr\'{a}l' proved that Οfβ(G)β€32/11β2.909. In this paper,
we prove that Οfβ(G)β€43/15β2.867