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), subset of > is a chain-complete and non-atomic partial order of the size 2(aleph 0). A family A subset of R(R) is a maximal antichain in this partial order iff (1) A boolean AND B does not contain a copy of R, for each different A, B is an element of A and (2) For each S is an element of R(R) there is A is an element of A such that A boolean AND S contains a copy of R. We show that the partial order < R(R), subset of > contains maximal antichains of size 2(aleph 0), aleph(0) 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 <[omega](omega), subset of >.