## Abstract

Given a graph *H*, the extremal number ex(*n, H*) is the largest number of edges in an *H*-free graph on *n* vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph *K*_{s,t}, we show that \({\rm{ex}}(n,K_{s,t}^\prime ) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for *t* sufficiently large in terms of *s*. Second, for any integers *s,k* ≥ 1, we show that \({\rm{ex}}(n,L) = \Theta ({n^{1 + {s \over <Stack><Subscript>+ 1</Subscript></Stack>}}})\) for a particular graph *L* depending on *s* and *k*, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number *r* ∈ (1, 2) is realisable in the sense that ex(*n, H*) = *Θ*(*n*^{r}) for some appropriate graph *H*, giving infinitely many new realisable exponents and implying that 1 + 1/*k* is a limit point of realisable exponents for all *k* ≥ 1. Writing *H*^{k} for the *k*-subdivision of a graph *H*, this result also implies that for any bipartite graph *H* and any *k*, there exists *δ* > 0 such that ex(*n, H*^{k−1}) = *O*(*n*^{1+1/k−δ}), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph *H* with maximum degree r on one side which does not contain *C*_{4} as a subgraph satisfies ex(*n, H*) = *o*(*n*^{2−1/r}).

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## Acknowledgements

We would like to thank the anonymous referees for their careful reviews.

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## Additional information

Research supported by ERC Starting Grant RanDM 676632.

Research supported by ERC Consolidator Grant PEPCo 724903.

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Conlon, D., Lee, J. & Janzer, O. More on the Extremal Number of Subdivisions.
*Combinatorica* **41, **465–494 (2021). https://doi.org/10.1007/s00493-020-4202-1

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### Mathematics Subject Classification (2010)

- 05C35