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