Switch to: References

Add citations

You must login to add citations.
  1. Complicated colorings, revisited.Assaf Rinot & Jing Zhang - 2023 - Annals of Pure and Applied Logic 174 (4):103243.
    Download  
     
    Export citation  
     
    Bookmark  
  • Strong colorings over partitions.William Chen-Mertens, Menachem Kojman & Juris Steprāns - 2021 - Bulletin of Symbolic Logic 27 (1):67-90.
    A strong coloring on a cardinal $\kappa $ is a function $f:[\kappa ]^2\to \kappa $ such that for every $A\subseteq \kappa $ of full size $\kappa $, every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction [A]^2$. The symbol $$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$ asserts the existence of a strong coloring on $\kappa $.We introduce the symbol $$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$ which asserts the existence of a coloring $f:[\kappa ]^2\to \kappa $ which is strong over a partition $p:[\kappa ]^2\to (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • New combinatorial principle on singular cardinals and normal ideals.Toshimichi Usuba - 2018 - Mathematical Logic Quarterly 64 (4-5):395-408.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • An extension of Shelah’s trichotomy theorem.Shehzad Ahmed - 2019 - Archive for Mathematical Logic 58 (1-2):137-153.
    Shelah develops the theory of \\) without the assumption that \\), going so far as to get generators for every \\) under some assumptions on I. Our main theorem is that we can also generalize Shelah’s trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for \\) which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.
    Download  
     
    Export citation  
     
    Bookmark  
  • 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.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Chain conditions of products, and weakly compact cardinals.Assaf Rinot - 2014 - Bulletin of Symbolic Logic 20 (3):293-314,.
    The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis of (...)
    Download  
     
    Export citation  
     
    Bookmark   14 citations  
  • 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.
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • 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 $\theta < \mu$.
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • 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 $\mu$.
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • 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.
    Download  
     
    Export citation  
     
    Bookmark   1 citation