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.
Categories: math.CO math.NT