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free Souslin tree Strong coloring Ulam matrix Erdos-Hajnal graphs Foundations Selective Ultrafilter stick Rainbow sets Partition Relations Uniformly coherent HOD Postprocessing function Amenable C-sequence transformations Well-behaved magma Almost-disjoint family Almost Souslin Minimal Walks incompactness Universal Sequences Strongly Luzin set super-Souslin tree 54G20 Fast club Axiom R Open Access Ramsey theory over partitions Hedetniemi's conjecture Parameterized proxy principle club_AD Dowker space AIM forcing Cardinal Invariants Singular cofinality O-space free Boolean algebra Subadditive Large Cardinals Martin's Axiom higher Baire space Chromatic number Prevalent singular cardinals Mandelbrot set Constructible Universe Prikry-type forcing C-sequence tensor product graph Uniformly homogeneous S-Space reflection principles Fodor-type reflection stationary hitting ZFC construction Small forcing Analytic sets Club Guessing Successor of Singular Cardinal Rado's conjecture Chang's conjecture square square principles Shelah's Strong Hypothesis Hindman's Theorem Generalized Clubs weak Kurepa tree weak diamond Non-saturation diamond star Antichain Diamond for trees specializable Souslin tree Filter reflection Precaliber polarized partition relation Weakly compact cardinal Kurepa Hypothesis Whitehead Problem Subnormal ideal Poset GMA middle diamond Forcing Forcing Axioms Nonspecial tree Reduced Power Sierpinski's onto mapping principle Subtle tree property OCA stationary reflection Generalized descriptive set theory regressive Souslin tree Ineffable cardinal Hereditarily Lindelöf space Greatly Mahlo sap ccc Sakurai's Bell inequality Singular Density Souslin Tree xbox Diamond-sharp Sigma-Prikry Fat stationary set Rock n' Roll P-Ideal Dichotomy Cohen real Jonsson cardinal unbounded function strongly bounded groups Iterated forcing Local Club Condensation. Square-Brackets Partition Relations Was Ulam right Diamond Dushnik-Miller Reflecting stationary set coloring number Ascent Path Ostaszewski square Microscopic Approach Uniformization Subtle cardinal PFA(S)[S] PFA Absoluteness Lipschitz reduction nonmeager set Commutative cancellative semigroups SNR very good scale weak square approachability ideal Singular cardinals combinatorics full tree Slim tree Knaster and friends indecomposable ultrafilter L-space b-scale Almost countably chromatic projective Boolean algebra Erdos Cardinal Successor of Regular Cardinal Closed coloring countably metacompact Aronszajn tree Distributive tree Cardinal function Knaster positive partition relation Coherent tree Vanishing levels Luzin set
Tag Archives: stationary reflection
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
The 15th International Workshop on Set Theory in Luminy, September 2019
I gave an invited talk at the 15th International Workshop on Set Theory in Luminy in Marseille, September 2019. Talk Title: Chain conditions, unbounded colorings and the C-sequence spectrum. Abstract: The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, … Continue reading
Posted in Invited Talks
Tagged Closed coloring, Knaster, Precaliber, stationary reflection, unbounded function
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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 remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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MFO workshop in Set Theory, February 2017
I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and … Continue reading
Posted in Invited Talks
Tagged Chromatic number, coloring number, incompactness, stationary reflection
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The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
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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Young Researchers in Set Theory, March 2011
These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … Continue reading