## Abstract

Abstract: A hypergraph H=(V,E) has property B_{k} if there exists a 2-coloring of the set V such that each edge contains at least k vertices of each color. We let m_{k, g}(n) and mΔ_{(k,b)}(n), respectively, denote the least number of edges of an n-homogeneous hypergraph without property BΔ_{k} which contains either no cycles of length at least g or no two edges intersecting in more than b vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for m^{*}_{k}(n), i.e., for the least number of edges of an n-homogeneous simple hypergraph without property B_{k}. Let Δ(H) be the maximal degree of vertices of a hypergraph H. By Δ_{k(n,g)} we denote the minimal degree Δ such that there exists an n-homogeneous hypergraph H with maximal degree Δ and girth at least g but without property B_{k}. In the paper, an upper bound for Δ_{k(n,g)} is obtained.

Original language | English (US) |
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Pages (from-to) | 188-200 |

Number of pages | 13 |

Journal | Mathematical Notes |

Volume | 108 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1 2020 |

### Bibliographical note

Funding Information:This work was supported by the Russian Foundation for Basic Research under grant 18-01-00355 and by the program “Leading Scientific Schools” (project no. NSh-2540.2020.1).

Publisher Copyright:

© 2020, Pleiades Publishing, Ltd.

## Keywords

- girth
- hypergraphs
- property B
- simple hypergraphs

## ASJC Scopus subject areas

- General Mathematics