## Abstract

We extend the notion of tournaments to orientation of the (k − 1)-skeleton of an (n − 1)-dimensional simplex. We ask for the maximal number of k-simplices whose boundary matches the orientation, extending the question on the upper bound of the number of directed 3-cycles of a tournament. In the general case we show polynomial upper and lower bounds. For the case k = 3 we improve the lower bound. Furthermore, this lower bound reaches the upper bound when n or n − 1 is a prime power congruent to 3 modulo 4.

Original language | English |
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Pages (from-to) | 209-236 |

Number of pages | 28 |

Journal | Combinatorica |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2021 |

### Bibliographical note

Publisher Copyright:© 2021, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

### Funding

The authors thank David Leep for the proof of Lemma 4.4 and Peter Sarnak for directing us to the cross ratio. We thank Fernando Xuancheng Shao for pointing us to results about “primes in short intervals”. We also thank Margaret Readdy and the two referees for their comments on an earlier version of this paper. This work was partially supported by a grant from the Simons Foundation (#429370 to Richard Ehrenborg).

Funders | Funder number |
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Fernando Xuancheng Shao | |

Margaret Readdy | |

Simons Foundation | 429370 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics