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Souslin Tree GMA Local Club Condensation. Knaster Vanishing levels Sigma-Prikry Uniformly coherent positive partition relation Uniformization Constructible Universe Strongly Luzin set xbox Erdos-Hajnal graphs strongly bounded groups Diamond specializable Souslin tree Aronszajn tree Commutative cancellative semigroups Non-saturation Diamond for trees Martin's Axiom unbounded function Precaliber regressive Souslin tree Reduced Power Hereditarily Lindelöf space Partition Relations Club Guessing Uniformly homogeneous Almost Souslin Selective Ultrafilter ccc C-sequence Hedetniemi's conjecture transformations stationary reflection P-Ideal Dichotomy b-scale higher Baire space L-space full tree Dowker space Dushnik-Miller Foundations Singular Density S-Space Chromatic number Small forcing OCA Coherent tree Whitehead Problem Well-behaved magma Kurepa Hypothesis diamond star Parameterized proxy principle Subtle cardinal Cardinal Invariants free Boolean algebra Rainbow sets Fodor-type reflection Forcing Axioms nonmeager set Was Ulam right Slim tree very good scale Shelah's Strong Hypothesis weak square Ostaszewski square square stick Weakly compact cardinal Fat stationary set Open Access Square-Brackets Partition Relations Amenable C-sequence Almost countably chromatic approachability ideal Fast club Closed coloring Cohen real Successor of Regular Cardinal projective Boolean algebra ZFC construction Knaster and friends Forcing Ineffable cardinal indecomposable ultrafilter Strong coloring Microscopic Approach Hindman's Theorem Axiom R reflection principles Chang's conjecture PFA polarized partition relation Generalized descriptive set theory Rock n' Roll PFA(S)[S] Jonsson cardinal Prikry-type forcing Sakurai's Bell inequality HOD AIM forcing sap Singular cofinality 54G20 Prevalent singular cardinals Postprocessing function Ascent Path Reflecting stationary set Absoluteness Lipschitz reduction stationary hitting Antichain Almost-disjoint family Diamond-sharp middle diamond Filter reflection O-space Subadditive Cardinal function Subnormal ideal Nonspecial tree incompactness coloring number Rado's conjecture Ramsey theory over partitions super-Souslin tree Distributive tree countably metacompact Minimal Walks SNR Luzin set Poset weak diamond Greatly Mahlo Analytic sets Mandelbrot set square principles Successor of Singular Cardinal Erdos Cardinal Universal Sequences Subtle tree property tensor product graph Large Cardinals Iterated forcing club_AD Generalized Clubs free Souslin tree Singular cardinals combinatorics Ulam matrix Sierpinski's onto mapping principle
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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