### Abstract

We say (n, e) → (m, f), an (m, f) subgraph is forced, if every n-vertex graph of size e has an m-vertex spanned subgraph with f edges. For example, as Turán proved, (n, e) → (k, (^{k}_{2})) for e > t_{k-1} (n) and (n, e) → (k, (^{k}_{2})), otherwise. We give a number of constructions showing that forced pairs are rare. Using tools of extremal graph theory we also show infinitely many positive cases. Several problems remain open.

Original language | English |
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Pages (from-to) | 61-77 |

Number of pages | 17 |

Journal | Discrete Mathematics |

Volume | 200 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Apr 6 1999 |

### Keywords

- Density
- Ramsey's theorem
- Turán's theorem

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Erdos, P., Füredi, Z., Rothschild, B. L., & Sós, V. T. (1999). Induced subgraphs of given sizes.

*Discrete Mathematics*,*200*(1-3), 61-77. https://doi.org/10.1016/S0012-365X(98)00387-2