11 found
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  1.  11
    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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  2.  35
    Successors of Singular Cardinals and Coloring Theorems I.Todd Eisworth & Saharon Shelah - 2005 - Archive for Mathematical Logic 44 (5):597-618.
    .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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  3.  9
    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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  4.  35
    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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  5. Forcing and Stable Ordered–Union Ultrafilters.Todd Eisworth - 2002 - Journal of Symbolic Logic 67 (1):449-464.
    We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those P-points that survive such forcing, answering a question left open by Blass [4]. We investigate the question of when this variant of Matet forcing can be used to diagonalize small filters without destroying P-points in the ground model. We also deal with the question of generic existence of stable ordered-union ultrafilters.
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  6.  7
    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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  7.  27
    Simultaneous Reflection and Impossible Ideals.Todd Eisworth - 2012 - Journal of Symbolic Logic 77 (4):1325-1338.
    We prove that if ${\mu ^ + } \to \left[ {{\mu ^ + }} \right]_\mu ^2 + $ holds for a singular cardinal μ, then any collection of fewer than cf(μ) stationary subsets of μ⁺ must reflect simultaneously.
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  8.  12
    On Ideals Related to I[Λ].Todd Eisworth - 2005 - Notre Dame Journal of Formal Logic 46 (3):301-307.
    We describe a recipe for generating normal ideals on successors of singular cardinals. We show that these ideals are related to many weakenings of □ that have appeared in the literature. Our main purpose, however, is to provide an organized list of open questions related to these ideals.
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  9.  38
    Near Coherence and Filter Games.Todd Eisworth - 2001 - Archive for Mathematical Logic 40 (3):235-242.
    We investigate a two-player game involving pairs of filters on ω. Our results generalize a result of Shelah ([7] Chapter VI) dealing with applications of game theory in the study of ultrafilters.
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  10.  2
    On Idealized Versions of Pr1).Todd Eisworth - 2014 - Archive for Mathematical Logic 53 (7-8):809-824.
    We obtain an improvement of some coloring theorems from Eisworth :1216–1243, 2010), Eisworth and Shelah :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) that maximizes the indecomposability of the associated ideal.
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  11.  1
    Representability and compactness for pseudopowers.Todd Eisworth - 2022 - Archive for Mathematical Logic 61 (1):55-80.
    We prove a compactness theorem for pseudopower operations of the form \}\) where \\le {{\,\mathrm{cf}\,}}\). Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which \\) has an inaccessible accumulation point.
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