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