Let $\alpha(G)$ denote the independence number of a graph $G$. Prove that there is an absolute positive constant $C$ such that for any $n\geq 1$ there is a graph $G$ on $3n$ vertices satisfying $\alpha (G) \leq C\log_{2}{n}, \alpha (G^2) \leq C(\log_{2}{n})^2,$ and $\alpha(G^3) \geq n$. (We define graph products here by the strong product.)
I'm thinking that a way to tackling this problem is to construct a graph $G$ that satisfies those properties for general $n$ and find a bound for the $C$ to set as the absolute constant, but I'm not sure how to begin with constructing such graphs in the general case.