On the extremal number of subdivisions

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich, and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and Cn2−1/r edges contains a copy of H⁠. This result is tight up to the constant when H contains a copy of Kr,s with s sufficiently large in terms of r⁠. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r=2⁠. More precisely, we show that if H is a C4-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and Cn3/2−δ edges contains a copy of H⁠. This answers a question of Erd̋s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest