On solution-free sets of integers II

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.