Forcing-theoretic framework for the Fraïssé theory

The abstract of the dissertation Forcing-theoretic framework for the Fraïssé theory Ziemowit Kostana 15.03.2021 Subject of the dissertation The subject of the dissertation is the study of possible generalizations of the Fraïssé theory, using the method of forcing. In 1954 Roland Fraïssé discovered that many classes of finite models, like graphs or linear orders, can be canonically assigned certain infinite models. These infinite models are universal – they contain isomorphic copies of all finite models from the class – and homogeneous – each isomorphism between finite substructures can be extended to an automorphism of the whole structure. An infinite structure with these two properties is called a Fraïssé limit. This correspondence is reversible – given a countable, homogeneous model, one can recover the class of finite models from which it was built – it is exactly the class of its finite substructures. The Fraïssé theory studies this correspondence. The Fraïssé limit of a class K has a natural connection with the forcing Fn (!;K; !) = fA 2 Kj F(A) 2 [!] !, the generic structure is typically rigid, i.e. has no automorphisms other than the identity. By slightly modifying the forcing, we can add structures together with automorphisms. Recall, that a linear order is !1-dense if each nonempty open subset has size !1, and separable if has a countable dense subset. Theorem (Kostana). The following is consistent with ZFC: There exists an !1-dense, separable linear order (A; ) together with an automorphism : A ! A, such that • 8a 2 A (a) > a; • Aut (A; ) = f kj k 2 Zg: In particular Aut (A; ) ' (Z; +). Chapter 5 We continue the study of models from Chapter 4, focusing on the models of size !1. Applying the ideas of Avraham, Rubin, and Shelah from the 80s, we show that in many cases the generically added structures become homogeneous after suitably extending the model of set theory, so that it satisfies Martin’s Axiom. In this extended models, they also have certain Ramsey-like properties. Theorem (Avraham-Shelah). The following is consistent with ZFC +MA!1 : There exists an !1-dense separable linear order L with the property that each uncountable partial function f L L is monotone on an uncountable set. Theorem (Kostana). The following is consistent with ZFC +MA!1 : There exists a separable metric space X of size !1 with the property that each uncountable partial 1-1 function f X X is an isometry on an uncountable set. Moreover, distances between points of the space X are rational, and X has a dense copy of the rational Urysohn space. 2