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countably metacompact Forcing PFA Analytic sets Slim tree Singular Density ZFC construction Strongly Luzin set Cohen real Distributive tree Almost-disjoint family Prikry-type forcing indecomposable ultrafilter Ramsey theory over partitions Shelah's Strong Hypothesis Cardinal Invariants incompactness Coherent tree free Boolean algebra Parameterized proxy principle Rado's conjecture Club Guessing HOD Constructible Universe nonmeager set strongly bounded groups Strong coloring Subtle tree property approachability ideal Prevalent singular cardinals AIM forcing sap reflection principles Rock n' Roll Erdos Cardinal C-sequence Almost Souslin Universal Sequences P-Ideal Dichotomy O-space Non-saturation Closed coloring very good scale Hindman's Theorem Foundations Forcing Axioms Chang's conjecture higher Baire space Filter reflection projective Boolean algebra Antichain Knaster Ineffable cardinal tensor product graph Dowker space Mandelbrot set b-scale Poset Dushnik-Miller Minimal Walks Sigma-Prikry Was Ulam right PFA(S)[S] Erdos-Hajnal graphs Chromatic number Commutative cancellative semigroups unbounded function Kurepa Hypothesis GMA club_AD transformations Precaliber Cardinal function Well-behaved magma Iterated forcing Uniformization Diamond for trees Nonspecial tree diamond star Ostaszewski square Whitehead Problem Hereditarily Lindelöf space 54G20 weak square Generalized descriptive set theory Ulam matrix Reduced Power Uniformly coherent Subadditive Absoluteness L-space specializable Souslin tree weak diamond Singular cardinals combinatorics coloring number stick Almost countably chromatic Square-Brackets Partition Relations Ascent Path full tree Greatly Mahlo Luzin set Axiom R Weakly compact cardinal super-Souslin tree S-Space ccc Uniformly homogeneous Microscopic Approach Rainbow sets positive partition relation Large Cardinals Lipschitz reduction Amenable C-sequence Jonsson cardinal xbox OCA SNR Subtle cardinal polarized partition relation Sierpinski's onto mapping principle middle diamond Sakurai's Bell inequality Knaster and friends free Souslin tree stationary reflection stationary hitting Diamond Selective Ultrafilter Aronszajn tree Successor of Regular Cardinal Successor of Singular Cardinal Reflecting stationary set Diamond-sharp Fast club Generalized Clubs Partition Relations Local Club Condensation. Fodor-type reflection Singular cofinality Subnormal ideal Hedetniemi's conjecture Fat stationary set Vanishing levels Small forcing Open Access Postprocessing function regressive Souslin tree Martin's Axiom square principles square Souslin Tree
Blog Archives
Squares, ultrafilters and forcing axioms
Joint work with Chris Lambie-Hanson and Jing Zhang. Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following. Strong forcing axioms, in general incompatible with the existence of indexed squares, can … Continue reading
Posted in Compactness, Preprints
Tagged Forcing Axioms, indecomposable ultrafilter, Subadditive, unbounded function
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Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … Continue reading
Posted in Partition Relations, Publications
Tagged Ascent Path, Knaster and friends, Open Access, P-Ideal Dichotomy, sap, square, Subadditive, Uniformly coherent
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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
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 forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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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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