Forcing with copies of the Rado and Henson graphs

Abstract

If B is a relational structure, define P(B) the partial order of all substructures of B that are isomorphic to it. Improving a result of Kurilić and the second author, we prove that if R is the random graph, then P(R) is forcing equivalent to S⁎R˙, where S is Sacks forcing and R˙ is an ω-distributive forcing that is not forcing equivalent to a σ-closed one. We also prove that P(H3) is forcing equivalent to a σ-closed forcing, where H3 is the generic triangle-free graph.