Strong Odd Colorings in Graph Classes of Bounded Expansion

We prove that for every d N and every graph class of bounded expansion C, there exists some c N so that every graph from C admits a proper coloring with at most c colors satisfying the following condition: in every ball of radius d, every color appears either zero times or an odd number of times. For d=1, this provides a positive answer to a question raised by Goetze, Klute, Knauer, Parada, Peña, and Ueckerdt ArXiv 2505. 02736 about the boundedness of the strong odd chromatic number in graph classes of bounded expansion. The key technical ingredient towards the result is a proof that the strong odd coloring number of a sets system can be bounded in terms of its semi-ladder index, 2VC dimension, and the maximum subchromatic number among induced subsystems.