# On the complexity of finding and counting solution-free sets of integers

## TL;DR

This paper explores the computational complexity of identifying and counting solution-free subsets of integers with respect to linear equations, connecting combinatorial number theory with algorithmic complexity.

## Contribution

It introduces the study of parameterized complexity for problems involving $ ext{L}$-free sets, including decision and counting problems, and raises open questions.

## Key findings

- Deciding the existence of large $ ext{L}$-free subsets is analyzed.
- Counting all $ ext{L}$-free subsets is examined.
- Open problems in the complexity of these problems are proposed.

## Abstract

Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving $\mathcal{L}$-free sets of integers. The main questions we consider involve deciding whether a finite set of integers $A$ has an $\mathcal{L}$-free subset of a given size, and counting all such $\mathcal{L}$-free subsets. We also raise a number of open problems.