  We study set systems formed by neighborhoods in graphs of bounded twin-width.
We start by proving that such graphs have linear neighborhood complexity, in
analogy to previous results concerning graphs from classes with bounded
expansion and of bounded clique-width. Next, we shift our attention to the
notions of distality and abstract cell decomposition, which come from model
theory. We give a direct combinatorial proof that the edge relation is distal
in classes of ordered graphs of bounded twin-width. This allows us to apply
Distal cutting lemma and Distal regularity lemma, so we obtain powerful
combinatorial tools for graphs of bounded twin-width.
