Almost-linear neighborhood complexity for monadically dependent classes

Establish that every monadically dependent graph class has almost-linear neighborhood complexity: for every ε>0, prove that the growth function of the neighborhood system of every graph in the class is bounded by b(ε,𝒞)n^{1+ε}, where b(ε,𝒞) is a constant depending only on ε and the class 𝒞.

Background

The paper explains that bounded VC-dimension yields a polynomial bound on the growth function, also called neighborhood complexity, via the Sauer–Shelah lemma. It then discusses stronger conjectured bounds for more restricted model-theoretic graph classes.

The unresolved claim concerns monadically dependent classes and predicts an almost-linear growth-function bound. The paper contrasts this conjecture with the known result for monadically stable classes and the linear neighborhood-complexity bound known for classes of bounded twin-width.