Weak coloring numbers for monadically stable classes

Develop a robust analogue of weak coloring numbers for monadically stable graph classes.

Background

Weak coloring numbers provide the ordering-based machinery used to construct sparse neighborhood covers in nowhere dense classes. The survey explains that this approach is unavailable for monadically stable classes because no comparable invariant is currently known.

References

This approach is currently not available in the monadically stable case, due to the lack of a robust analogue of weak coloring numbers; developing such an analogue is a notorious open problem.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Section 4.2, paragraph following Theorem 4.4