
TL;DR
This paper explores new variations of model-theoretic tree properties, establishing equivalences and a dichotomy theorem, and introduces the $ ext{c}$-tree property related to instability and indiscernibles.
Contribution
It introduces the $ ext{c}$-tree property and its variants, proving their equivalences to known properties and establishing a dichotomy theorem for these properties.
Findings
$ ext{c}$-TP is equivalent to instability.
$ ext{c}$-$ ext{TP}_i$ is equivalent to $ ext{TP}_i$.
$ ext{c}$-$ ext{TP}_{ii}$ is equivalent to IP.
Abstract
In this paper, we introduce novel variations on several well-known model-theoretic tree properties, and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the -tree property (-TP), for an arbitrary Ramsey index structure . We focus attention on the colored linear order index structure \textbf{c}, showing that \textbf{c}-TP is equivalent to instability. After introducing \textbf{c}- and \textbf{c}-, we prove that \textbf{c}- is equivalent to , and that \textbf{c}- is equivalent to IP. We see that these three tree properties give a dichotomy theorem, just as with TP, , and . Along the way, we observe that appropriately generalized tree index structures are Ramsey, allowing for the use of generalized tree indiscernibles.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Graph Theory Research · Limits and Structures in Graph Theory
