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