Classify attainable asymptotic growth functions

Determine the families of functions n↦n(n) for which there exist a graph G and a graph family F satisfying lim n→∞ φ(n,G|F)/n(n)=1, where φ is either f or g when defined.

Background

The authors introduce a general asymptotic classification problem for the possible growth rates of f and g. Their notation distinguishes the sets of asymptotically attainable functions for the two invariants.