VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width

We study set systems formed by neighborhoods in graphs of bounded twin-width. In particular, we prove that such classes of graphs admit linear neighborhood complexity, in analogy to previous results concerning classes with bounded expansion and classes of bounded clique-width. Additionally, we show how, for a given graph from a class of graphs of bounded twin-width, to efficiently encode the neighborhood of a vertex in a given set of vertices $A$ of the graph. For the encoding we use only a constant number of vertices from $A$. The obtained encoding can be decoded using FO formulas. This proves that the edge relation in graphs of bounded twin-width, seen as first-order structures, admits a definable distal cell decomposition. From this fact we derive that we can apply to such classes combinatorial tools based on the Distal cutting lemma and the Distal regularity lemma