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  1. On idealized versions of Pr1(μ +, μ +, μ +, cf(μ)).Todd Eisworth - 2014 - Archive for Mathematical Logic 53 (7):809-824.
    We obtain an improvement of some coloring theorems from Eisworth (Fund Math 202:97–123, 2009; Ann Pure Appl Logic 161(10):1216–1243, 2010), Eisworth and Shelah (J Symb Logic 74(4):1287–1309, 2009) for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an “idealized” version of the combinatorial principle Pr1(μ +, μ +, μ +, cf(μ)) that maximizes the indecomposability of the associated ideal.
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  • Getting more colors II.Todd Eisworth - 2013 - Journal of Symbolic Logic 78 (1):17-38.
    We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,cf(\mu))$ for singular $\mu$.
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  • Colouring and non-productivity of ℵ2-C.C.Saharon Shelah - 1997 - Annals of Pure and Applied Logic 84 (2):153-174.
    We prove that colouring of pairs from 2 with strong properties exists. The easiest to state problem it solves is: there are two topological spaces with cellularity 1 whose product has cellularity 2; equivalently, we can speak of cellularity of Boolean algebras or of Boolean algebras satisfying the 2-c.c. whose product fails the 2-c.c. We also deal more with guessing of clubs.
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  • Getting more colors I.Todd Eisworth - 2013 - Journal of Symbolic Logic 78 (1):1-16.
    We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta < \mu$.
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  • Knaster and friends II: The C-sequence number.Chris Lambie-Hanson & Assaf Rinot - 2020 - Journal of Mathematical Logic 21 (1):2150002.
    Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong (...)
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  • Club-guessing, stationary reflection, and coloring theorems.Todd Eisworth - 2010 - Annals of Pure and Applied Logic 161 (10):1216-1243.
    We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and , then there is a regular cardinal θ<μ such that any fewer than cf stationary subsets of must reflect simultaneously.
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  • (1 other version)Successors of singular cardinals and coloring theorems I.Todd Eisworth & Saharon Shelah - 2005 - Archive for Mathematical Logic 44 (5):597-618.
    Abstract.We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [1], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
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  • Was sierpinski right? IV.Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (3):1031-1054.
    We prove for any $\mu = \mu^{ large enough (just strongly inaccessible Mahlo) the consistency of 2 μ = λ → [θ] 2 3 and even 2 μ = λ → [θ] 2 σ,2 for $\sigma . The new point is that possibly $\theta > \mu^+$.
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  • Was Sierpiński right? III Can continuum-c.c. times c.c.c. be continuum-c.c.?Saharon Shelah - 1996 - Annals of Pure and Applied Logic 78 (1-3):259-269.
    We prove the consistency of: if B 1 , B 2 are Boolean algebras satisfying the c.c.c. and the 2 ℵo -c.c. respectively then B 1 × B 2 satisfies the 2 ℵo -c.c. We start with a universe with a Ramsey cardinal.
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  • (1 other version)Successors of Singular Cardinals and Coloring Theorems II.Todd Eisworth & Saharon Shelah - 2009 - Journal of Symbolic Logic 74 (4):1287 - 1309.
    In this paper, we investigate the extent to which techniques used in [10], [2], and [3]—developed to prove coloring theorems at successors of singular cardinals of uncountable cofinality—can be extended to cover the countable cofinality case.
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