Rainbow matchings and algebras of sets

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Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number v=v(n) such that, if A1,..., An 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 Ai-equivalence classes of size larger than 1, then X has a rainbow matching-a set of 2n distinct elements a1, b1,..., an, bn, such that ai is Ai-equivalent to bi 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 languageEnglish
Pages (from-to)251-257
Number of pages7
JournalElectronic Notes in Discrete Mathematics
StatePublished - Nov 2015


  • Algebra of sets
  • Equivalence relation
  • Rainbow matching

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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