We prove that for every d N and a 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, Pe\~na, 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.
Michał Pilipczuk (Wed,) studied this question.