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