A note on first-order spectra with binary relations

The spectrum of a first-order sentence is the set of the cardinalities of its finite models. In this paper, we consider the spectra of sentences over binary relations that use at least three variables. We show that for every such sentence $\mathrm{\Phi }$, there is a sentence ${\mathrm{\Phi }}^{\mathrm{\prime }}$ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of $\mathrm{\Phi }$. Moreover, the models of ${\mathrm{\Phi }}^{\mathrm{\prime }}$ are all bipartite graphs. As a corollary, we obtain that to settle Asser’s conjecture, i.e., whether the class of spectra is closed under complement, it is sufficient to consider only sentences using only three variables whose models are restricted to undirected bipartite graphs.