## Abstract

Let H be a graph on h vertices. The number of induced copies of H in a graph G is denoted by i _{H} (G). Let i _{H} (n) denote the maximum of i _{H} (G) taken over all graphs G with n vertices. Let f(n,h)=Π _{i} ^{h} a _{i} where ∑ _{i=1} ^{h} a _{i} =n and the a _{i} are as equal as possible. Let g(n,h)=f(n,h)+∑ _{i=1} ^{h} g(a _{i} ,h). It is proved that for almost all graphs H on h vertices it holds that i _{H} (n)=g(n,h) for all n≤2 ^{h} . More precisely, we define an explicit graph property P _{h} which, when satisfied by H, guarantees that i _{H} (n)=g(n,h) for all n≤2 ^{h} . It is proved, in particular, that a random graph on h vertices satisfies P _{h} with probability 1−o _{h} (1). Furthermore, all extremal n-vertex graphs yielding i _{H} (n) in the aforementioned range are determined. We also prove a stability result. For H∈P _{h} and a graph G with n≤2 ^{h} vertices satisfying i _{H} (G)≥f(n,h), it must be that G is obtained from a balanced blowup of H by adding some edges inside the blowup parts. The inducibility of H is i _{H} =lim _{n→∞} i _{H} (n)/(nh). It is known that i _{H} ≥h!/(h ^{h} −h) for all graphs H and that a random graph H satisfies almost surely that i _{H} ≤h ^{3logh} h!/(h ^{h} −h). We improve upon this upper bound almost matching the lower bound. It is shown that a graph H which satisfies P _{h} has i _{H} =(1+O(h ^{ −h 1/3 } ))h!/(h ^{h} −h).

Original language | English |
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Pages (from-to) | 81-109 |

Number of pages | 29 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 136 |

DOIs | |

State | Published - May 2019 |

### Bibliographical note

Funding Information:This research was supported in part by the Israel Science Foundation (grant No. 1082/16).

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Induced density
- Inducibility
- Packing

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics