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Coherent forests


Author: Monroe Eskew
Journal: Proc. Amer. Math. Soc. 143 (2015), 2705-2717
MSC (2010): Primary 03E05, 03E10, 03E20, 03E35, 03E55
DOI: https://doi.org/10.1090/S0002-9939-2015-12622-6
Published electronically: February 4, 2015
MathSciNet review: 3326048
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Abstract: A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ZFC. We give several ways of obtaining coherent Suslin forests by forcing, one of which generalizes the well-known argument of Todorčević that a Cohen real adds a Suslin tree. Another uses a strong combinatorial principle that plays a similar role to diamond. We show that, starting from a large cardinal, this principle can be obtained by a forcing that is small relative to the forest it constructs.


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Monroe Eskew
Affiliation: International Research Fellow of the Japan Society for the Promotion of Science. Department of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan

DOI: https://doi.org/10.1090/S0002-9939-2015-12622-6
Received by editor(s): November 21, 2013
Published electronically: February 4, 2015
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society