On solution-free sets of integers II

TL;DR

This paper precisely determines the maximum size of solution-free sets for certain linear equations and bounds the number of maximal such sets, extending results using Green's container and removal lemmas.

Contribution

It provides exact sizes of largest solution-free sets and bounds on the count of maximal solution-free sets for classes of linear equations of the form px+qy=rz.

Findings

Exact maximum size of $ ext{L}$-free subsets for several classes of linear equations.

Upper bounds on the number of maximal $ ext{L}$-free subsets, tight in some cases.

Extension of results to linear equations with more than three variables.

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 $\mathcal{L}$. We determine the precise size of the largest $\mathcal{L}$-free subset of $[n]$ for several general classes of linear equations $\mathcal{L}$ of the form $px+qy=rz$ for fixed $p,q,r \in \mathbb N$ where $p \geq q \geq r$. Further, for all such linear equations $\mathcal{L}$, we give an upper bound on the number of maximal $\mathcal{L}$-free subsets of $[n]$. In the case when $p=q\geq 2$ and $r=1$ this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green to prove this result. Our results also extend to various linear equations with more than three variables.