## Abstract

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G|→∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (Ars Combin 22:5–10, 1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (Combinatorica 17(1):125–134, 1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

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

Pages (from-to) | 1367-1380 |

Number of pages | 14 |

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 5 |

DOIs | |

State | Published - 24 Sep 2015 |

### Bibliographical note

Publisher Copyright:© 2014, Springer Japan.

## Keywords

- Graph density
- Inducibility
- Ramsey multiplicity