Subdivisions with congruence constraints in digraphs of large chromatic number.

We prove that for every digraph F $F$ and every assignment of pairs of integers (re,qe)e∈A(F) ${({r}_{e},{q}_{e})}_{e\in A(F)}$ to its arcs there exists an integer N $N$ such that every digraph D $D$ with dichromatic number greater than N $N$ contains a subdivision of F $F$ in which e $e$ is subdivided into a directed path of length congruent to re ${r}_{e}$ modulo qe ${q}_{e}$, for every e∈A(F) $e\in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result. [ABSTRACT FROM AUTHOR]