Possible spectra of a fixed sequence

Characterize the sets of positive integers that can occur as the S-spectrum of a graph for a fixed sequence S.

Background

For an infinite sequence S, the S-spectrum of a graph consists of the numbers of colors with which the graph realizes the corresponding initial segment of S. The parameters χ_S(G) and φ_S(G) record the minimum and maximum elements of this spectrum.

The paper notes that, beyond cases forced by regular-graph upper bounds, the behavior of these spectra and associated extremal parameters is largely unexplored. The open question asks which sets of positive integers are realizable as spectra.