### Abstract

Some inequalities for cross-unions of families of finite sets are proved that are related to the problem of minimizing the union-closure of a uniform family of given size. The cross-union of two families F and G of subsets of [n]={1,2,…,n}[n]={1,2,…,n} is the family F∨G={F∪G:F∈F,G∈G}. It is shown that |F∨G|/|F|≥|G∨B

_{n}|/2^{n}|, where B_{n}denotes the power set of [n]. Besides, the problem of minimizing |F∨G|over all union-closed FF and G∨G generated by a given number*r*of singletons and a given number s > (r 2) of two-sets, respectively, is solved.Original language | English |
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Pages (from-to) | 392-401 |

Number of pages | 10 |

Journal | European Journal of Combinatorics |

Volume | 35 |

DOIs | |

Publication status | Published - 2014 |

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## Cite this

Leck, U., & Roberts, I. (2014). Inequalities for cross-unions of collections of finite sets.

*European Journal of Combinatorics*,*35*, 392-401. https://doi.org/10.1016/j.ejc.2013.06.036