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We prove a half-space theorem for an ideal Scherk graph $\Sigma\subsetM\times\mathbb R$ over a polygonal domain $D\subset M,$ where $M$ is a Hadamardsurface whose curvature is bounded above by a negative constant. Moreprecisely, we show that a properly immersed minimal surface contained in$D\times\mathbb R$ and disjoint from $\Sigma$ is a translate of $\Sigma.$

A half-space theorem for ideal Scherk graphs in $M\times\mathbb R$