### Abstract

A finite set system (hypergraph)H is said to have the Helly property if the members of each intersecting subsystem H′ of H share an element (i.e. H∩H′≠øFS for all H,H′∈H′ implies ∩_{H∈H′} H≠ø); H is k-uniform if |H|=k for all H∈H. In a previous paper, we con jectured that the union of t k-uniform Helly families (k≥3) on the same n-element set can have at most ∑^{t} _{i=1}(^{n-i} _{k-1}) members. Here we prove this conjecture for every k and every t ≤2k-2, for n sufficiently large with respect to k. The main tool is a result (an analogue of the Hilton-Milner theorem on intersecting k-uniform set systems) stating that if the sets in a k-uniform Helly family on n points have an empty intersection, then for large n, |H|≤(^{n-k-1} _{k-1})+(^{n-2} _{k-2})+1 , and the set system attaining equality is uniq ue. We also show that ∑^{t} _{i=1}(^{n-i} _{k-1}) is a (best possible) upper bound for the union of t intersecting k-uniform set systems on the same n-element set, whenever n≥(t+2k-1)(k-1). This result strengthens a particular case of the Hajnal-Rothschild theorem.

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

Pages (from-to) | 319-327 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 127 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Mar 15 1994 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

## Fingerprint Dive into the research topics of 'Largest size and union of Helly families'. Together they form a unique fingerprint.

## Cite this

*Discrete Mathematics*,

*127*(1-3), 319-327. https://doi.org/10.1016/0012-365X(92)00488-D