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~$d$.

翻译：我们给出了$\nu(d,k)$的本质上紧的界，其中$\nu(d,k)$表示在双宽至多为$d$的图中，集合$X$（包含$k$个顶点）上不同邻域的最大数目。利用经典的Marcus-Tardos定理，两项独立工作[Bonnet等，Algorithmica '22; Przybyszewski '22]证明了上界$\nu(d,k) \leqslant \exp(\exp(O(d)))k$，其双宽依赖性为双指数形式。而[Gajarsky等，ICALP '22]的工作利用局部类型框架，暗示了单指数界的存在（但未明确给出该界）。我们给出了这样一个显式界，并证明其本质上是紧的。具体而言，我们提供了一个简短的自包含证明：对于任意$d$和$k$，有$\nu(d,k) \leqslant (d+2)2^{d+1}k = 2^{d+\log d+\Theta(1)}k$；同时，在$k$相对于$d$足够大的情况下，我们构造了一个二分图，表明$\nu(d,k) \geqslant 2^{d+\log d+\Theta(1)}k$。