Triangle-free graph with sub-power independence number

For every sufficiently large $n$, there is a triangle-free graph $G$ on $n$ vertices with $\alpha (G)<n^{0.7}$. One construction samples $G(2n,(2n{)}^{-0.69})$: with positive probability it has fewer than $n$ triangles and no independent set of size $n^{0.7}$. Delete one vertex from each triangle and then take an induced $n$-vertex subgraph.