Erdős–Simonovits Zarankiewicz comparison conjecture

Prove that for every bipartite graph H, the asymmetric Zarankiewicz number satisfies Z(n,n,H)=\Theta(ex(n,n,H)), where ex(n,n,H) is the maximum number of edges in an H-free subgraph of K_{n,n}.

Background

The Zarankiewicz number Z(n,n,H) permits an H-free subgraph of K_{n,n} with the prescribed bipartition, whereas ex(n,n,H) denotes the corresponding balanced bipartite extremal number without the prescribed placement of the parts of H.

The paper establishes that this conjecture is equivalent to both vertex-gluing conjectures above. This equivalence explains why the seemingly simple graph-gluing question is connected to a classical unresolved problem in extremal graph theory.