## Abstract

Grinblat (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)=3n-2 for all n≥. 4. In this paper we improve the upper bound (for all large enough n) to v(n)≤16n/5+O(1).

Original language | English |
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Pages (from-to) | 251-257 |

Number of pages | 7 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 49 |

DOIs | |

State | Published - Nov 2015 |

## Keywords

- Algebra of sets
- Equivalence relation
- Rainbow matching

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics