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