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Indivisible sets and well‐founded orientations of the Rado graph
Author(s) -
Ackerman Nathanael L.,
Brian Will
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600057
Subject(s) - countable set , mathematics , combinatorics , discrete mathematics , indecomposable module
Abstract Every set can been thought of as a directed graph whose edge relation is ∈ . We show that many natural examples of directed graphs of this kind are indivisible: H κ for every infinite κ, V λ for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find 2 ℵ 0indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ 1 that are distinct up to siblinghood.