## Abstract

Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that D(n,T) = 2^{tk-1(n)} for all sufficiently (very) large n, where t _{k-1}(n) is the maximum possible number of edges of a graphon n vertices with no K _{k} , (determined by Turán's Theorem). The proof is based on a directed version of Szemerédi's regularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity lemma and therefore show that in these cases D(n,T) = 2[^{n2/4]} already holds for (relatively) small values of n.

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

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2006 |

### Bibliographical note

Funding Information:* Research supported in part by a USA Israeli BSF grant, by a grant from th e Israel Science Foundation and by th e Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics