The spectrum of a first-order logic sentence is the set of natural numbers
that are cardinalities of its finite models. In this paper we study the
hierarchy of first-order spectra based on the number of variables. It has been
conjectured that it collapses to three variable. We show the opposite: it forms
an infinite hierarchy. However, despite the fact that more variables can
express more spectra, we show that to establish whether the class of
first-order spectra is closed under complement, it is sufficient to consider
sentences using only three variables and binary relations.Comment: 13 page
