Halin's grid theorem for digraphs

Abstract: Halin showed that every thick end of every graph contains an infinite grid. We extend Halin's theorem to digraphs. More precisely, we show that for every infinite family $\mathcal{R}$ of disjoint equivalent out-rays there is a grid whose vertical rays are contained in $\mathcal{R}$. Furthermore, we obtain similar results for in-rays and necklaces.