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Intersection theorems for systems of sets (III)

Published online by Cambridge University Press:  09 April 2009

P. Erdös ,
E. C. Milner  and
R. Rado
P. Erdös
Affiliation:
University of Reading, England
E. C. Milner
Affiliation:
University of Reading, England
R. Rado
Affiliation:
University of Calgary, Canada
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A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 1 , August 1974 , pp. 22 - 40
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Erdös, P. and Rado, R., ‘Intersection theorems for systems of sets’, J. London Math. Soc. 35 (1960), 85–90.CrossRefGoogle Scholar
[2]Erdös, P. and Rado, R., ‘Intersection theorems for systems of sets (II)’, J. London Math. Soc. 44 (1969), 467–479.CrossRefGoogle Scholar
[3]Erdös, P., Hajnal, A. and Rado, R., ‘Partition relations for cardinal numbers’, Acta Math. Acad. Scient. Hungarica 16 (1965), 93–196.CrossRefGoogle Scholar