Isometric copies of directed trees in orientations of graphs
Banakh, Taras, 1968- ...
The isometric Ramsey number ▫$\mathsf{IR} (\overrightarrow{\mathcal{H}})$▫ of a family ▫$\overrightarrow{\mathcal{H}}$▫ of digraphs is the smallest number of vertices in a graph ▫$G$▫ such that any ...orientation of the edges of ▫$G$▫ contains every member of ▫$\overrightarrow{\mathcal{H}}$▫ in the distance-preserving way. We observe that the isometric Ramsey number of a finite family of finite acyclic digraphs is always finite, and present some bounds in special cases. For example, we show that the isometric Ramsey number of the family of all oriented trees with ▫$n$▫ vertices is at most ▫$n^{2 n + o (n)}$▫.
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