## Abstract

A family of sets is called r-cover free if no set in the family is contained in the union of r (or less) other sets in the family. A 1-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's classical result determines the maximal cardinality of a 1-cover free family of subsets of an n-element set. Estimating the maximal cardinality of an r-cover free family of subsets of an n-element set for r>1 was also studied. In this note we are interested in the following probabilistic variant of this problem. Let S_{0},S_{1},…,S_{r} be independent and identically distributed random subsets of an n-element set. Which distribution minimizes the probability that S_{0}⊆⋃_{i=1}^{r}S_{i}? A natural candidate is the uniform distribution on an r-cover-free family of maximal cardinality. We show that for r=1 such distribution is indeed best possible. In a complete contrast, we also show that this is far from being true for every r>1 and n large enough.

Original language | English |
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Article number | 112027 |

Journal | Discrete Mathematics |

Volume | 343 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2020 |

### Bibliographical note

Funding Information:This research was partially supported by National Science Foundation, USA Grant DMS-1855464 , U.S.-Israel Binational Science Foundation, USA Grant 2018267 , the Simons Foundation, USA , a Google Research Award, NSF-BSF, USA Grant 2018640 , the Israel Science Foundation Grant 1018/16 , and the Center for Cyber Law and Policy at the University of Haifa, Israel , in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office.

Funding Information:

This research was partially supported by National Science Foundation, USA Grant DMS-1855464, U.S.-Israel Binational Science Foundation, USA Grant 2018267, the Simons Foundation, USA, a Google Research Award, NSF-BSF, USA Grant 2018640, the Israel Science Foundation Grant 1018/16, and the Center for Cyber Law and Policy at the University of Haifa, Israel, in conjunction with the Israel National Cyber Bureau in the Prime Minister's Office.

Publisher Copyright:

© 2020 Elsevier B.V.

## Keywords

- Cover free families
- Sperner's theorem