### Abstract

We consider the structure of H-free subgraphs of graphs with high minimal degree. We prove that for every k>m there exists an ϵ≔ϵ(k,m)>0 so that the following holds. For every graph H with chromatic number k from which one can delete an edge and reduce the chromatic number, and for every graph G on n>n_{0}(H) vertices in which all degrees are at least (1−ϵ)n, any subgraph of G which is H-free and contains the maximum number of copies of the complete graph K_{m} is (k−1)-colorable. We also consider several extensions for the case of a general forbidden graph H of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs.

Original language | English (US) |
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Pages (from-to) | 988-996 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 342 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2019 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*342*(4), 988-996. https://doi.org/10.1016/j.disc.2018.11.012