## 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 \(\mathfrak {v} = \mathfrak {v}(n)\) such that, if \(A_1, \ldots , A_n\) are *n* equivalence relations on a common finite ground set *X*, such that for each *i* there are at least \(\mathfrak {v}\) elements of *X* that belong to \(A_i\)-equivalence classes of size larger than 1, then *X* has a rainbow matching—a set of 2*n* distinct elements \(a_1, b_1, \ldots , a_n, b_n\), such that \(a_i\) is \(A_i\)-equivalent to \(b_i\) for each *i*? Grinblat has shown that \(\mathfrak {v}(n) \le 10n/3 + O(\sqrt{n})\). He asks whether \(\mathfrak {v}(n) = 3n-2\) for all \(n\ge 4\). In this paper we improve the upper bound (for all large enough *n*) to \(\mathfrak {v}(n) \le 16n/5 + O(1)\).

## Keywords

Equivalence Class Equivalence Relation Pairwise Disjoint Distinct Element Combinatorial Problem## Notes

### Acknowledgements

Thanks to L. Š. (Yehuda) Grinblat for suggesting us to look at this problem and for helpful discussions. Special thanks to the referees for reading the paper carefully and providing detailed suggestions. Thanks also to Anat Paskin-Cherniavsky for helpful discussions.

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