Spectra of high n and non-low n degrees

We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of non-low n degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order ℚ. New results include realizations of the set of non-low n Turing degrees as the spectrum of a relation on ℚ for all n≥1, and a realization of the set of all non-low n Turing degrees as the spectrum of a linear order whenever n≥2. The state of current knowledge is summarized in a table in the concluding section. © 2010 The Author. Published by Oxford University Press. All rights reserved