Neighbourhood complexity of graphs of bounded twin-width

Abstract

We give essentially tight bounds for, ν(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 ν(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 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 kν(d,k)⩽(d+2)2d+1k=2d+logd+Θ(1)k,and build a bipartite graph implying ν(d,k)⩾2d+logd+Θ(1)k, in the regime when k is large enough compared to d.