Every infinite family of disjoint equivalent directed rays in a digraph contains a directed quarter-grid with those rays as vertical rays, with an analogous result for necklace-based ends.
Halin, ¨Uber die Maximalzahl fremder unendlicher Wege in Graphen, Mathematische Nachrichten30 (1965), no
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Halin's grid theorem for digraphs
Every infinite family of disjoint equivalent directed rays in a digraph contains a directed quarter-grid with those rays as vertical rays, with an analogous result for necklace-based ends.