Indiscernibles, EM-Types, and Ramsey Classes of Trees

Abstract

Abstract. It was shown in [16] that for a certain class of structures I, Iindexed indiscernible sets have the modeling property just in case the age of I is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we obtain a new Ramsey class of finite trees.

Ramsey transfer to semi-retractions

“…Then we apply compactness to the type S where we replace T ∀ ∪ Diag(I) with the diagram of I in L ′′ . It was noted in the proof for [12,Thm 3.12] that the qfi hypothesis was used only in the argument for Claim 3.13, and in this Remark we point out why it is not even needed there. In terms of the compactness argument presented, the more important assumption is that structures A, B being described in the type S are finite, and can be listed in finitely many variables.…”

“…It is often desirable to find an I-indexed indiscernible set that witnesses specific definable configurations in U. For this, we define the notion of EM-type: 12]). Given an L ′ -structure I, an L-structure U and an I-indexed set of same-length tuples from U, X = (a i | i ∈ I), we define the Ehrenfeucht-Mostowski type (EM-type) of X to be a syntactic type in variables (x i | i ∈ I) such that for ψ(x 1 , .…”

“…Theorem 2.7. [ [12,Thm 3.12]] Suppose that I is a qfi, locally finite structure in a language L ′ with a relation < linearly ordering I. Then I-indexed indiscernible sets have the modeling property if and only if age(I) has RP.…”

“…In fact, it was later pointed out to the author that the qfi assumption is not needed (see Acknowledgements). To see this, in the argument for [12,Claim 3.13] we replace L ′ with expansion L ′′ that contains predicates p A (x) for all complete quantifier-free types of finite substructures A of I. Then we apply compactness to the type S where we replace T ∀ ∪ Diag(I) with the diagram of I in L ′′ .…”

On the Antichain Tree Property

“…We also recall some knowledge of Ramsey class. The following definitions and notations are from [17], [22], and [28]. Definition 2.6.…”