On model-theoretic tree properties

Chernikov, Artem; Ramsey, Nicholas

doi:10.1142/s0219061316500094

Cited by 63 publications

(124 citation statements)

References 18 publications

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“…By a very slight modification of the proof of [, Lemma 4.6], we get: Proposition Suppose

{false(a_{η} false)}_{η \in 2^{< ω}}

is a tree strongly indiscernible over C such that

(a_{0^{α}} : 0 < α < ω)

is indiscernible over

c C

. Let

p false(y; z false) = tp false(c; {false(a_{0^{⌢} 0^{γ}} false)}_{γ < ω} false) / C false)

, and let

q (y)

be a partial type over

⌀

contained in tp( c ).…”

Section: Overview Of Some Results On the Sop2 By Chernikov And Ramseymentioning

confidence: 99%

“…By compactness and indiscernibility, there is a formula

ψ \in p

such that

q false(y false) \cup {ψ false(y, a_{0}, \dots, a_{0^{⌢} 0^{n - 1}} false), ψ false(y, a_{1}, \dots, a_{1^{⌢} 0^{n - 1}} false)}

is inconsistent. Then as in , the n ‐fold elongation (cf. [, Definition 2.6]) of

{false(a_{η} false)}_{η \in 2^{< ω}}

witnesses that ψ has the SOP 2 inside of q .

□

…”

Section: Overview Of Some Results On the Sop2 By Chernikov And Ramseymentioning

confidence: 99%

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A preservation theorem for theories without the tree property of the first kind

Dobrowolski

Kim

2017

Mathematical Logic Qtrly

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We prove that the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H-structures associated to T . We also provide a class of examples of nonsimple geometric NTP 1 theories. IntroductionOne theme of research in model theory is to inquire whether some well-known properties are preserved under a certain unary predicate expansions of a given structure. One of the motivations for this is that positive theorems of this kind often allows us to obtain interesting and complicatedlooking theories which still satisfy some strong tameness conditions. The study of expansions by unary predicates reaches back to the paper of Poizat on beautiful pairs [10], and has been developed in various directions ever since. Remarkable papers on this subject include for example [5], where the stability condition is examined, and [1], which is dedicated to studying expansions in simple theories (which generalize the stable ones).The well-known equivalence TP ⇔ TP 1 ∨ TP 2 due to Shelah [11] (where TP denotes the tree property while TP 1 and TP 2 denote the tree properties of the first and second kind, respectively), suggests two natural generalizations of simple theories, namely NTP 1 theories and NTP 2 theories (i.e., theories without TP 1 and TP 2 , respectively). So far, NTP 1 and NTP 2 theories have been studied much less extensively than the simple ones (i.e. theories without TP). However, recently some interesting results on these theories began to appear, notably [6] and [7]. In particular, natural examples of non-simple NTP 1 thoeries were provided in [6], namely: ω-free PAC fields, linear spaces with a generic bilinear form and a class of theories obtained by the "pfc" construction.The study of expansions in the NTP 2 context was undertaken in [2], where it was shown that the NTP 2 property is preserved under "dense and codense" unary predicate expansions of geometric structures where the unary predicate is assumed to define either an algebraically independent subset or an elementary substructure. In the present paper, we prove that the NTP 1 property is also preserved under such expansions. One of the main ingredients in our proof is the recently proved fact (due to Chernikov and Ramsey [6]) that the TP 1 property can, in any TP 1 theory, always be witnessed by some formula in a single free variable. We also prove (in Section 4) that an NTP 1 nonsimple geometric theory can be obtained from any Fraïssé limit which has a simple *

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“…By a very slight modification of the proof of [, Lemma 4.6], we get: Proposition Suppose

{false(a_{η} false)}_{η \in 2^{< ω}}

is a tree strongly indiscernible over C such that

(a_{0^{α}} : 0 < α < ω)

is indiscernible over

c C

. Let

p false(y; z false) = tp false(c; {false(a_{0^{⌢} 0^{γ}} false)}_{γ < ω} false) / C false)

, and let

q (y)

be a partial type over

⌀

contained in tp( c ).…”

Section: Overview Of Some Results On the Sop2 By Chernikov And Ramseymentioning

confidence: 99%

“…By compactness and indiscernibility, there is a formula

ψ \in p

such that

q false(y false) \cup {ψ false(y, a_{0}, \dots, a_{0^{⌢} 0^{n - 1}} false), ψ false(y, a_{1}, \dots, a_{1^{⌢} 0^{n - 1}} false)}

is inconsistent. Then as in , the n ‐fold elongation (cf. [, Definition 2.6]) of

{false(a_{η} false)}_{η \in 2^{< ω}}

witnesses that ψ has the SOP 2 inside of q .

□

…”

Section: Overview Of Some Results On the Sop2 By Chernikov And Ramseymentioning

confidence: 99%

A preservation theorem for theories without the tree property of the first kind

Dobrowolski

Kim

2017

Mathematical Logic Qtrly

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show abstract

“…Then, we define: Finally, we will outline the "pfc" construction from Subsection 6.3 of [4]. For the reader's convenience, we repeat the definitions used there.…”

Section: Definition 210 ([1]) a Suitable Tuple Is A Sequencementioning

confidence: 99%

“…2, we review some definitions and outline the generic variations construction from [1] as well as the pfc construction from [4]. We notice that the theory obtained in the second construction is interpretable in the theory obtained in the first one.…”

Section: Introductionmentioning

confidence: 99%