In this paper, we study the notions related to tree property 1 (=TP1), or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in Džamonja and Shelah (2004) is crucially used throughout the paper.
Bibliographical noteFunding Information:
The first author was supported by NRF grant 2010-0016044. The second author was supported by Hi Seoul Science/Humanities Fellowship from Seoul Scholarship Foundation, and by the Brain Korea 21 Program 2006–2009. This paper is a part of the second author’s Ph.D. thesis.
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