Raphael Steiner, Congruence-constrained subdivisions in digraphs

I will present the short proof from [1] that for every digraph F and every assignment of pairs of integers $(r_e,q_e{)}_{e\in A(F)}$ to its arcs, there exists an integer $N$ such that every digraph D with dichromatic number at least $N$ 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 $e\in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs and at the same time yields a novel proof of his result. I will also talk about how a hypergraph coloring result from [2] may help to obtain good bounds on $N$ in the special case when $F$ is subcubic.

[1] https://arxiv.org/abs/2208.06358

[2] https://arxiv.org/abs/2206.13635