This paper will examine universality spectra for relational theories which cannot be described in first-order logic. We will give a method using functors to show that two types of structures have the same universality spectrum. A combination of methods will be used to show universality results for certain ordered structures and graphs. In some cases, a universal spectrum under GCH will be obtained. Since the theories are not first-order, the classic model theory result under GCH does not hold.
"Universality for Orders and Graphs Which Omit Large Substructures." Notre Dame J. Formal Logic 47 (2) 233 - 248, 2006. https://doi.org/10.1305/ndjfl/1153858648