
> 
> A fractional coloring of $G(V,E)$ is a function $f:{2^{|V|}} \to [0,1]$ such that $f(S) > 0$
>  only if $S$ is an independent set in $G$, and for all $v \in V$, it holds $\sum\nolimits\_{S:v \in S} {f(S) \ge 1} $. A fractional chromatic number $\chi\_f(G)$ is the minimum $k$ for which there exists a fractional coloring $f$ for $G$ with $\sum\nolimits\_S {f(S) = k} $.
> 
> 
> Let $G = (V,E) $ be a graph and let $k \in \mathbb{N}$. Show that
>  ${\chi\_f}(G) \le k$ if and only if for all weight function $w:V \to
>  \mathbb{R}\_+$ there is an independent set of size at least $\left(
>  {\sum\nolimits\_{v \in V} {w(v)} } \right)/k$. Moreover, if we are
>  given an algorithm that computes an independent set of size $w(V)/k$
>  for all weight function $w$, then this algorithm can be used to
>  compute a fractional coloring using $(1 + \varepsilon )k$ colors.
> 
> 
> 


Where i can read about the proof of this claim?



Fractional Coloring
[A Note on Fractional Coloring and the Integrality gap of LP for Maximum Weight Independent Set](http://www.sciencedirect.com/science/article/pii/S1571065316301858), Parinya Chalermsook and Daniel Vaz (2016).


