## 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 Kst′ for the subdivision of the bipartite graph K_{s,t}, we show that ex(nKst′)=O(n3/2−12s). 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 ex(nL)=Θ(n1+ssk+1) 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}).

Original language | English |
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Pages (from-to) | 465-494 |

Number of pages | 30 |

Journal | Combinatorica |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2021 Aug |

### Bibliographical note

Funding Information:Research supported by ERC Consolidator Grant PEPCo 724903. Acknowledgements

Funding Information:

Research supported by ERC Starting Grant RanDM 676632.

Publisher Copyright:

© 2021, János Bolyai Mathematical Society and Springer-Verlag.

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics