
Abstract
Writing $\mathcal{I}(G)$ for the collection of independent sets of a given graph $G$, a random independent set $\mathbf{I}$ drawn according to the hardcore distribution at fugacity $\lambda$ on $\mathcal{I}(G)$ satisfies that
$\Pr[\mathbf{I} = I] = \dfrac{\lambda^{I}}{Z_\lambda(G)}$,
for every independent set $I\in \mathcal{I}(G)$, where
$\displaystyle Z_\lambda(G) = \sum_{I\in \mathcal{I}(G)} \lambda^{I}$
is a normalising factor, called the independence polynomial of $G$.
In 2002, Molloy and Reed used the uniform distribution over the independent sets of maximum size in a graph together with a greedy fractional colouring algorithm to prove that the Reed conjecture holds for the fractional chromatic number, namely $\chi_f(G) \le (\omega(G)+\Delta(G)+1)/2$ for every graph $G$. Note that the hardcore distribution at fugacity $\lambda=\infty$ corresponds to the uniform distribution on the independent sets of maximum size. We generalise their framework by using the hardcore distribution in various settings, in order to prove several fractional colouring results on sparse graphs, most of which match with the best known bounds in the offdiagonal Ramsey theory. As a bonus, all those results can be stated in a local fashion, so that each vertex $v \in V(G)$ may be coloured by a subset included in the interval $[0, \gamma(v)]$, where $\gamma(v)$ depends on the degree of $v$.
Main theorem
Let $\varepsilon>0$ be some fixed real. For every graph $G$, there exists a fractional colouring $w$ of $G$ such that
 if $G$ is of girth at least $7$, then
$\displaystyle \forall v\in V(G), \quad w(v) \subseteq \left[0, \min_{k \in \mathbb{N}_{\ge 4}} \dfrac{2\deg(v)+2^{k3}+k}{k}\right]$;
 if $G$ is trianglefree, then there exists $\delta_{\varepsilon}$ such that
$\displaystyle \forall v\in V(G), \quad w(v) \subseteq \left[0, (1+\varepsilon) \max \left\{\dfrac{\deg(v)}{\ln \deg(v)}, \dfrac{\delta_{\varepsilon}}{\ln \delta_{\varepsilon}} \right\}\right]$;
 if every vertex of $G$ is contained in at most $T>0$ triangles, then there exists $\delta_{\varepsilon,T}$ such that
$\displaystyle \forall v\in V(G), \quad w(v) \subseteq \left[0, (1+\varepsilon) \max \left\{\dfrac{\deg(v)}{\ln \frac{\deg(v)}{\sqrt{T}}}, \dfrac{\delta_{\varepsilon,T}}{\ln \frac{\delta_{\varepsilon,T}}{\sqrt{T}}} \right\}\right]$.
