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A Criterion for the Existence of Transversals of Set Systems
Author(s) -
Wojciechowski Jerzy
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005528
Subject(s) - uncountable set , countable set , transversal (combinatorics) , mathematics , set (abstract data type) , discrete mathematics , combinatorics , computer science , mathematical analysis , programming language
Nash‐Williams [ 6 ] formulated a condition that is necessary and sufficient for a countable family A=( A i ) i ∈ I of sets to have a transversal. In [ 7 ] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that ∩ i ∈ J A i =Ø for any uncountable J ⊆ I . In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash‐Williams for any family u. We also present a self‐contained proof that if ∩ i ∈ J A i =Ø for any uncountable J ⊆ I , then our condition is necessary and sufficient for the family u to have a transversal.