  Let $G$ be any triangle-free graph with maximum degree $\Delta\leq 3$. Staton
proved that the independence number of $G$ is at least 5/14n. Heckman and
Thomas conjectured that Staton's result can be strengthened into a bound on the
fractional chromatic number of $G$, namely $\chi_f(G)\leq 14/5. Recently,
Hatami and Zhu proved $\chi_f(G) \leq 3 -{3/64}$. In this paper, we prove
$\chi_f(G) \leq 3- 3/43$.
