## Abstract

Given any integers

*s*,*t*⩾*2*, we show that there exists some*c*=*c*(*s*,*t*)>*0*such that any*K*_{s,t}‐free graph with average degree*d*contains a subdivision of a clique with at least*cd**s*^{/}*2*^{(}*s*^{−}^{1}^{)}vertices. In particular, when*s*=*2*, this resolves in a strong sense the conjecture of Mader in 1999 that every*C*‐free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of_{4}*K*_{s,t }‐free graphs suggests our result is tight up to the constant*c*(*s*,*t*).Original language | English |
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Pages (from-to) | 203-222 |

Number of pages | 20 |

Journal | Journal of the London Mathematical Society |

Volume | 95 |

Issue number | 1 |

Early online date | 6 Jan 2017 |

DOIs | |

Publication status | Published - Feb 2017 |

## Keywords

- 05C35 (primary)
- 05C38
- 05C83 (secondary)

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