## Abstract

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich, and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n2 - 1/r edges contains a copy of H. This result is tight up to the constant when H contains a copy of Kr,s with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi's result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C4-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and Cn3/2-δ edges contains a copy of H. This answers a question of Erd¨s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.

Original language | English |
---|---|

Pages (from-to) | 9122-9145 |

Number of pages | 24 |

Journal | International Mathematics Research Notices |

Volume | 2021 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2021 Jun 1 |

### Bibliographical note

Funding Information:This work was supported by a Royal Society University Research Fellowship and by the European Research Council under Starting Grant 676632.

Publisher Copyright:

© 2019 The Author(s).

## All Science Journal Classification (ASJC) codes

- Mathematics(all)