Hierarchy theorem for approximation ratios? As is well known, NP-hard optimization problems can have many different approximation ratios, ranging all the way from having a PTAS to not being approximable within any factor. In between, we have various constants, $O(\log n)$, $poly(n)$, etc.

What is known about the set of possible ratios? Can we prove any sort of "approximation hierarchy"? Formally, for what functions $f(n)$ and $g(n)$ can we prove that there exists a problem with approximation ratio $f(n)\leq \alpha &lt; g(n)$?

In the case that $\alpha=O(1)$, does there exist a problem with approximation ratio exactly $\alpha$?
There is an approximation hierarchy, the main known examples: FPTAS $\subseteq$ EPTAS $\subseteq$ PTAS $\subseteq$ APX. But for inapproximability there also is NPO-PB.

There are a lot of results about the set of possible ratios, going from results like this one: 


  $P||C_{max} \in$ EPTAS$\backslash$ FPTAS, unless $P=NP$, 


to defining APX/NPO-PB-hard problems.

Some references:


ON PTAS: M. Cesati and L. Trevisan. On the efficiency of polynomial time approximation schemes, 1997. 
On NPOPB: V. Kann. Strong lower bounds on the approximability of some NPO PB-complete maximization problems


But I suggest the best will be to check the Complexity Zoo because it has many more information and references on those examples, even Wikipedia

Furthermore, as stated in the comments the tight bound when $\alpha=O(1)$, was shown for many problems such as bin packing, machine scheduling (see iris.gmu.edu/~khoffman/papers/set_covering.html).