Lemma1. There exists a polynomial time computable transformation $f$ from the 3CNF formulas to graphs such that for every 3CNF formula $\varphi$, $f(\varphi)$ is an $n$-vertex graph whose $|MIS| = \mathcal{V}(\varphi).\frac{n}{3}.$
(We denote $\mathcal{V}(\varphi)$ as the maximum fraction of the clauses satisfied over all assignments of values to the boolean variables.)
Lemma2. There exists a constant $\rho>1$ such that INDSET, the problem of independent set, cannot have a $\rho$-factor approximation algorithm,unless P = NP.
Proof of Lemma2. Let $L\in \textbf{NP}$. We know that the decision problem of L can be reduced to approximating MAX3SAT. The reduction produces an instance $\varphi$ of MAX3SAT where $\mathcal{V}(\varphi)=1$ ($\varphi$ is satisfied) or $\mathcal{V}(\varphi) < \frac{1}{\rho}$ where $\rho > 1$ is some constant. Apply the reduction of the earlier lemma1 to $\varphi$ to conclude that a $\rho$-factor approximation algorithm gives a $\rho$-factor approximation algorithm for MAX3SAT on $\varphi$. So, $\rho$-factor approximation algorithm for INDSET is NP-hard.
I am interested to prove this theorem:For every $\rho>1$, computing a $\rho$-factor approximation algorithm for INDSET is not possible, unless P = NP.
Maybe We have to amplify the approximation gap using graph product. But how to start and proceed?
