Maximal antichains of isomorphic subgraphs of the Rado graph

Abstract

© 2015, University of Nis. All rights reserved. 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 2Ȣ0 . A family A ⊂ R(R) is a maximal antichain in this partial order I (1) A∩B does not contain a copy of R, for each diffrent A; B ϵ A and (2) For each S ϵ R(R) there is A ϵ A such that A ∩ S contains a copy of R. We show that the partial order 〈R(R),⊂〉 contains maximal antichains of size 2Ȣ0, Ȣ0 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 〈[w]w;⊂〉.