TY - JOUR

T1 - Rainbow Matchings and Algebras of Sets

AU - Nivasch, Gabriel

AU - Omri, Eran

N1 - Publisher Copyright:
© 2017, Springer Japan.

PY - 2017/1

Y1 - 2017/1

N2 - 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 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) = 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).

AB - 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 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) = 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).

UR - http://www.scopus.com/inward/record.url?scp=85008441302&partnerID=8YFLogxK

U2 - 10.1007/s00373-017-1764-9

DO - 10.1007/s00373-017-1764-9

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85008441302

SN - 0911-0119

VL - 33

SP - 473

EP - 484

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 2

ER -