On model-theoretic tree properties

2015/05/03 by Chernikov, Artem, Ramsey, Nicholas · 3 citations
#03C45 #05C05 #05C55 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1505.00454

Abstract

We study model theoretic tree properties (TP, TP₁, TP₂) and their associated cardinal invariants (κ_cdt, κ_sct, κ_inp, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem (TP ⇒ TP₁ ∨ TP₂) for countable theories, show that TP₁ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak k-TP₁ is equivalent to TP₁ (answering a question of Kim and Kim). Besides, we give a characterization of NSOP₁ via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed NSOP₁.

Cited by

• Abstract independence relations in neostability theory
• A note on stable Kim-forking
• Results on Colored Tree Properties

Related

• Invariants Related to the Tree Property
• More on tree properties
• Bi-invariant types, reliably invariant types, and the comb tree property
• Theories without the tree property of the second kind
• Mekler's construction and tree properties