Locally sparse graph independence bound

There is an absolute $c>0$ such that if an $n$-vertex graph has maximum degree at most $d$ and every vertex neighbourhood spans at most $d^2/f$ edges, where $2\le f\le d^2$, thenThe triangle-free case is commonly called Shearer's independence bound.

Every triangle-free graph on $n$ vertices with maximum degree at most $d$ hasfor an absolute constant $c>0$.