A note on sublinear separators and expansion

For a hereditary class C of graphs, let s_C(n) be the minimum function such that each n-vertex graph in C has a balanced separator of order at most s_C(n), and let nabla_C(r) be the minimum function bounding the expansion of C, in the sense of bounded expansion theory of Nešetřil and Ossona de Mendez. The results of Plotkin, Rao, and Smith (1994) and Esperet and Raymond (2018) imply that if s_C(n)=Theta(n^{1-epsilon}) for some epsilon>0, then nabla_C(r)=Omega(r^{1/(2.epsilon)-1}/polylog r) and nabla_C(r)=O(r^{1/epsilon-1}polylog r). Answering a question of Esperet and Raymond, we show that neither of the exponents can be substantially improved.