Extremal density for subdivisions with length or sparsity constraints

Given a graph H , a balanced subdivision of H is obtained by replacing all edges of H with internally disjoint paths of the same length. In this paper, we prove that for any graph H , a linear-in- e ( H ) bound on average degree guarantees a balanced H -subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever H is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs H below this density, its subdivisions are forcible by a sublinear-in- e ( H ) bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph H with sublogarithmic density are forcible by a sublinear-in- e ( H ) bound on average degree, provided that H satisfies some additional separability condition.