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SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 4 / December 2018
- Published online by Cambridge University Press:
- 21 December 2018, pp. 1512-1538
- Print publication:
- December 2018
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- Article
- Export citation
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With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle
$\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of 
$\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals 
$\lambda < \kappa$, the principle 
$\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which 
${\aleph _2}$-Aronszajn trees exist and all such trees contain 
${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of 
$\square \left( \kappa \right)$.
