# A note on first-order spectra with binary relations

## TL;DR

This paper demonstrates that spectra of certain first-order sentences with binary relations can be represented by simpler sentences with a single symmetric relation, focusing on bipartite graphs, which simplifies the study of spectra closure properties.

## Contribution

It shows that spectra of sentences with at least three variables over binary relations can be reduced to spectra of sentences with one symmetric relation, models being bipartite graphs.

## Key findings

- Spectra are linearly proportional under the reduction.
- Models can be restricted to bipartite graphs.
- Implication for Asser's conjecture on spectra closure.

## Abstract

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 $\Phi$, there is a sentence $\Phi'$ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of $\Phi$. Moreover, the models of $\Phi'$ 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.