Subdivisions with congruence constraints in digraphs of large chromatic number

Abstract

We prove that for every digraph $F$ and every assignment of pairs of integers ${\left(r_e,q_e\right)}_{e\in A\left(F\right)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number greater than $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e\in A\left(F\right)$. 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.