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