Følner sequences and sum-free sets

Abstract: Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$ is a multiplicative F{\o}lner sequence in $\mathbf{N}$ then $F_m$ has no $k$-sum-free subset of size greater than $(1/(k+1)+o(1))|F_m|$. This provides a new proof and a generalisation of a recent theorem of Eberhard, Green, and Manners.