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., 2022; Przybyszewski, 2022) have shown the upper bound nu(d, k) <= exp(exp(O(d)))k, with a double-exponential dependence in the twin-width. The work of Gajarsky et al. (2022), using the frame-work 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) <= (d + 2)2(d+1)k = 2(d+logd+Theta(1))k,and build a bipartite graph implying nu(d, k) >= 2(d+logd+Theta(1)k), in the regime when k is large enough compared to d.(c) 2023 The Authors. Published by Elsevier Ltd.