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