# Fifty Years of the Spectrum Problem: Survey and New Results

## TL;DR

This paper surveys 50 years of research on the spectrum problem in logic, covering developments from number theory to graph theory, and presents new results and open questions in the field.

## Contribution

It provides a comprehensive overview of the spectrum problem's evolution and introduces new insights and unresolved questions in finite model theory and descriptive complexity.

## Key findings

- Scholz's question has been answered in various ways.
- Asser's question about spectra complements remains open.
- The survey highlights connections between logic, complexity, and graph theory.

## Abstract

In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. G\"unter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. One appendix paraphrases the contents of several early and not easily accesible papers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendix contains a compendium of questions and conjectures which remain open.