Occupancy fraction, fractional colouring, and triangle fraction

Given $varepsilon>0$, there exists $f_0$ such that, if $f_0 le f le Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $Delta$ in which the neighbourhood of every vertex in $G$ spans at most $Delta^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-varepsilon)(n/Delta)log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+varepsilon)Delta/log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as strong