Maximal Antichains of Isomorphic Subgraphs of the Rado Graph

If 〈R, E〉 is the Rado graph and R(R) the set of its copies inside R, then 〈R(R), ⊂〉 is a chain-complete and non-atomic partial order of the size 2 א 0 . A family A ⊂ R(R) is a maximal antichain in this partial order iff (1) A ∩ B does not contain a copy of R, for each different A, B ∈ A and (2) For...

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Bibliographic Details
Published in:Filomat 2015-01, Vol.29 (9), p.1919-1923
Main Authors: Kurilic, Milos, Markovic, Petar
Format: Article
Language:English
Subjects:
Cardinality
Integers
Mathematical set theory
Mathematics
Partially ordered sets
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Summary:If 〈R, E〉 is the Rado graph and R(R) the set of its copies inside R, then 〈R(R), ⊂〉 is a chain-complete and non-atomic partial order of the size 2 א 0 . A family A ⊂ R(R) is a maximal antichain in this partial order iff (1) A ∩ B does not contain a copy of R, for each different A, B ∈ A and (2) For each S ∈ R(R) there is A ∈ A such that A ∩ S contains a copy of R. We show that the partial order 〈R(R), ⊂〉 contains maximal antichains of size 2 א 0 , א0 and n, for each positive integer n (thus, of all possible cardinalities, under CH). The results are compared with the corresponding known results concerning the partial order 〈[ω]ω, ⊂〉.
ISSN:0354-5180
2406-0933
DOI:10.2298/FIL1509919K