### Abstract

For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . ., X_{k}) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Turán-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.

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

Pages (from-to) | 815-828 |

Number of pages | 14 |

Journal | Acta Cybernetica |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Keywords

- Crossing set
- Hypergraph
- Partition
- Set system
- Turán-type problem
- Upper chromatic number

### ASJC Scopus subject areas

- Theoretical Computer Science
- Software
- Computer Science (miscellaneous)
- Computer Vision and Pattern Recognition
- Management Science and Operations Research
- Information Systems and Management
- Electrical and Electronic Engineering

## Fingerprint Dive into the research topics of 'Partition-Crossing Hypergraphs'. Together they form a unique fingerprint.

## Cite this

*Acta Cybernetica*,

*23*(3), 815-828. https://doi.org/10.14232/actacyb.23.3.2018.6