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If $k$ is a positive real number, we say that a set $S$ of real numbers is $k$-sum-free if there do not exist $x,y,z$ in $S$ such that $x + y = kz$. For $k$ greater than or equal to 4 we find the essentially unique measurable $k$-sum-free subset of $(0,1]$ of maximum size.

Maximum subsets of (0,1] with no solutions to x+y = kz