If-direction of the 2-degenerate extremal bound

Prove that every 2-degenerate bipartite graph H satisfies ex(n,H)=O(n^{3/2}).

Background

The paper discusses the Erdős–Simonovits conjecture asserting that a bipartite graph has extremal number O(n{3/2}) if and only if it is 2-degenerate. The paper disproves the full biconditional by showing that the two directions cannot simultaneously hold, and notes that the converse direction had already been disproved by explicit constructions.

The unresolved part is the forward, or ‘if,’ direction: whether 2-degeneracy alone guarantees the O(n{3/2}) extremal bound. The paper constructs infinitely many examples with exactly one degree-2 vertex satisfying this bound, but does not prove the assertion for every 2-degenerate bipartite graph.

References

The if direction of~\cref{conj:r=2} remains a major open problem, even for the case $r = 2$: whether every $2$-degenerate bipartite $H$ satisfies $ex(n, H) = O(n{3/2})$.

— Bipartite Turán problem on graph gluing  (2501.12953 - Dong et al., 22 Jan 2025) in Section 1 (Introduction), paragraph following the disproof of Conjecture r