Neighbourhood complexity of graphs of bounded twin-width

We give essentially tight bounds for, $\nu(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ $k$ vertices in graph with twin-width at most~$d$. Using celebrated Marcus-Tardos theorem, two independent works [Bonnet et al., Algorithmica '22; Przybyszewski '22] have shown upper bound $\nu(d,k) \leqslant \exp(\exp(O(d)))k$, double-exponential dependence twin-width. The work [Gajarsky ICALP '22], using framework local types, implies existence single-exponential (without explicitly stating such bound). an explicit bound, and prove that it is tight. Indeed, we short self-contained proof for every $d$ $$\nu(d,k) (d+2)2^{d+1}k = 2^{d+\log d+\Theta(1)}k,$$ build bipartite implying \geqslant d+\Theta(1)}k$, regime when large enough compared to~$d$.