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