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