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