On solution-free sets of integers

TL;DR

This paper investigates the size, count, and structure of solution-free subsets of integers with respect to linear equations, providing exact and asymptotic results for specific classes of equations.

Contribution

It completely determines the maximum size of solution-free sets for equations of the form px+qy=z and refines bounds on their counts, using container and removal lemmas.

Findings

Exact size of largest solution-free sets for px+qy=z

Asymptotic count of solution-free subsets for broad classes of equations

Bounds on the number of maximal solution-free subsets for three-variable equations

Abstract

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. In this paper we consider the following three general questions: (i) What is the size of the largest $\mathcal{L}$-free subset of $[n]$? (ii) How many $\mathcal{L}$-free subsets of $[n]$ are there? (iii) How many maximal $\mathcal{L}$-free subsets of $[n]$ are there? We completely resolve (i) in the case when $\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\in \mathbb N$ where $p\geq 2$. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations $\mathcal{L}$, thereby refining a special case of a result of Green. We also give various bounds on the number of maximal $\mathcal{L}$-free subsets of $[n]$ for three-variable homogeneous linear equations $\mathcal{L}$. For this, we make use of container and removal lemmas of Green.