Type: Article

Forcing with copies of the Rado and Henson graphs

Guzmán, Osvaldo; Todorcevic, Stevo;

Abstract:

For a countable relational structure B let P(B) be the set of all substructures of B isomorphic to B. If A,C?P(B), define A?C if and only if A is a substructure of C. Let Q be the linear order of rational numbers, let R be the random graph, let H3 denote the 3-Henson graph (also known as the generic triangle-free graph), and let S be Sacks forcing. The starting points of the paper are the following results: (1) P(Q) is forcing equivalent to the iteration of S and a ?-closed forcing; (2) P(R) is forcing equivalent to an iteration of the form P?R? where P is a proper forcing that adds a real and has the 2-localization property and R? is a P-name of an ?-distributive forcing. In the paper under review the authors improve the latter result and prove that P(R) is forcing equivalent to S?R? where R? is an S-name of an ?-distributive forcing that is not forcing equivalent to a ?-closed forcing. They also prove that P(H3) is forcing equivalent to a ?-closed forcing.