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polarized partition relation Subtle tree property Coherent tree Subnormal ideal Uniformization Iterated forcing indecomposable ultrafilter Commutative cancellative semigroups Almost Souslin Diamond-sharp Local Club Condensation. Ramsey theory over partitions free Boolean algebra Subadditive Slim tree Uniformly homogeneous Hereditarily Lindelöf space Knaster and friends tensor product graph Precaliber HOD Reflecting stationary set Sierpinski's onto mapping principle weak Kurepa tree Analytic sets Fast club Constructible Universe transformations Lipschitz reduction Singular Density Greatly Mahlo reflection principles weak diamond approachability ideal weak square Knaster Successor of Singular Cardinal Universal Sequences Mandelbrot set Postprocessing function Uniformly coherent Was Ulam right Strongly Luzin set Sakurai's Bell inequality specializable Souslin tree Fat stationary set sap Jonsson cardinal countably metacompact square Vanishing levels Filter reflection Erdos Cardinal Almost-disjoint family Diamond for trees Rock n' Roll Cardinal Invariants C-sequence Whitehead Problem super-Souslin tree Generalized Clubs Poset Amenable C-sequence unbounded function Generalized descriptive set theory Absoluteness Prikry-type forcing regressive Souslin tree Erdos-Hajnal graphs Singular cardinals combinatorics Kurepa Hypothesis Weakly compact cardinal Non-saturation Club Guessing diamond star ccc Ascent Path Subtle cardinal Prevalent singular cardinals stick Dowker space Aronszajn tree 54G20 Rainbow sets Nonspecial tree square principles Microscopic Approach middle diamond PFA(S)[S] full tree Strong coloring incompactness Large Cardinals Souslin Tree projective Boolean algebra Martin's Axiom positive partition relation Fodor-type reflection Shelah's Strong Hypothesis Hindman's Theorem AIM forcing Dushnik-Miller Chromatic number Forcing Axioms club_AD Almost countably chromatic nonmeager set Well-behaved magma Ulam matrix stationary reflection Cardinal function GMA Rado's conjecture strongly bounded groups Partition Relations Parameterized proxy principle Reduced Power Singular cofinality Selective Ultrafilter L-space Diamond Hedetniemi's conjecture Ineffable cardinal coloring number Axiom R O-space stationary hitting b-scale Chang's conjecture Closed coloring SNR Luzin set Cohen real Antichain Distributive tree xbox Ostaszewski square Square-Brackets Partition Relations OCA PFA higher Baire space S-Space very good scale Forcing Foundations free Souslin tree Successor of Regular Cardinal ZFC construction Minimal Walks Open Access Sigma-Prikry Small forcing P-Ideal Dichotomy
Tag Archives: square
Complicated colorings, revisited
Joint work with Jing Zhang. Abstract. In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish that for every pair $\chi<\kappa$ of … Continue reading
Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … Continue reading
Posted in Partition Relations, Publications
Tagged Ascent Path, Knaster and friends, Open Access, P-Ideal Dichotomy, sap, square, Subadditive, Uniformly coherent
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Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
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Transformations of the transfinite plane
Joint work with Jing Zhang. Abstract. We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. 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 … Continue reading
11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
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Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … Continue reading
A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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The reflection principle $R_2$
A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading
Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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