Maximal antichains of isomorphic subgraphs of the Rado graph

If ?R,E? is the Rado graph andR(R) the set of its copies inside R, then ?R(R), ?? is a chain-complete and non-atomic partial order of the size 2x0 . 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 2x0, X0 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 ?[?]?, ??.