The following statement seems very plausible: If $G$ is a triangle-free graph of degeneracy $d$, then $$ \chi_f(G) \leq O(\frac{d}{\log d}) $$ where $\chi_f$ is the fractional chromatic number.
This statement would be false if $\chi_f$ is replaced with the ordinary chromatic number $\chi$; Alon, Krivelevich, Sudakov give a counter-example of a $d$-degenerate graph with chromatic number $d$. However, their counter-example also has fractional chromatic number $O(d/\log d)$.
I could not find any reference to this statement, either stating it as a known conjecture or giving a proof. Is this known anywhere?
