Our research addresses such questions. We identify a broad class of reductions -- namely, code-like reductions -- that includes all reductions used in known inapproximability results. We prove, under reasonable complexity assumptions, that code-like reductions cannot prove the hardness of approximating Clique within a factor $\sqrt{n}$, of approximating Vertex Cover within a factor 1.5, and of coloring a 3-colorable graph with $O(\log n)$ colors.
The above results help explain why it has been difficult to prove really good inapproximability result.
At the end, we will also argue that it is inherently difficult to come up with reductions that are not code-like. We provide some evidence supporting this view-point.
(The talk will be self-contained.)