## Abstract

Grinblat (Algebras of Sets and Combinatorics, Translations of Mathematical Monographs, vol. 214. AMS, Providence, 2002) asks the following question in the context of algebras of sets: What is the smallest number v= v(n) such that, if A_{1}, … , A_{n} are n equivalence relations on a common finite ground set X, such that for each i there are at least v elements of X that belong to A_{i}-equivalence classes of size larger than 1, then X has a rainbow matching—a set of 2n distinct elements a_{1}, b_{1}, … , a_{n}, b_{n}, such that a_{i} is A_{i}-equivalent to b_{i} for each i? Grinblat has shown that v(n)≤10n/3+O(n). He asks whether v(n) = 3 n- 2 for all n≥ 4. In this paper we improve the upper bound (for all large enough n) to v(n) ≤ 16 n/ 5 + O(1).

Original language | English |
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Pages (from-to) | 473-484 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Jan 2017 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics