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very good scale Strong coloring Generalized Clubs free Boolean algebra Jonsson cardinal Vanishing levels 54G20 Singular cardinals combinatorics Uniformization Analytic sets Forcing middle diamond Successor of Singular Cardinal Commutative cancellative semigroups Lipschitz reduction OCA Well-behaved magma square Filter reflection Hindman's Theorem Cardinal Invariants Foundations Minimal Walks reflection principles Whitehead Problem AIM forcing strongly bounded groups stationary reflection Rainbow sets SNR indecomposable ultrafilter Souslin Tree Singular Density Chang's conjecture Subnormal ideal Was Ulam right Parameterized proxy principle Aronszajn tree Diamond-sharp Postprocessing function GMA sap b-scale approachability ideal Coherent tree square principles Nonspecial tree Rock n' Roll super-Souslin tree full tree Poset Square-Brackets Partition Relations PFA Fodor-type reflection Diamond for trees Almost Souslin stick O-space Mandelbrot set Diamond Antichain Distributive tree Non-saturation HOD Subtle cardinal Sierpinski's onto mapping principle Uniformly homogeneous Almost-disjoint family Fast club PFA(S)[S] Erdos Cardinal Ramsey theory over partitions polarized partition relation transformations Ulam matrix Forcing Axioms Singular cofinality Club Guessing Reduced Power Axiom R projective Boolean algebra club_AD Weakly compact cardinal diamond star Precaliber Prevalent singular cardinals Generalized descriptive set theory Large Cardinals Microscopic Approach Hereditarily Lindelöf space Knaster and friends S-Space L-space Kurepa Hypothesis ZFC construction Almost countably chromatic Absoluteness positive partition relation weak diamond Rado's conjecture Sigma-Prikry Local Club Condensation. Universal Sequences specializable Souslin tree countably metacompact Partition Relations Ineffable cardinal regressive Souslin tree nonmeager set Fat stationary set Cardinal function tensor product graph higher Baire space Prikry-type forcing Uniformly coherent C-sequence Ascent Path Cohen real Luzin set Hedetniemi's conjecture Strongly Luzin set Amenable C-sequence Erdos-Hajnal graphs Dushnik-Miller Successor of Regular Cardinal Slim tree Dowker space weak Kurepa tree Greatly Mahlo Martin's Axiom stationary hitting xbox Ostaszewski square Sakurai's Bell inequality Subadditive Knaster Iterated forcing Selective Ultrafilter P-Ideal Dichotomy free Souslin tree Reflecting stationary set unbounded function Constructible Universe Closed coloring coloring number Small forcing Open Access Shelah's Strong Hypothesis weak square ccc Chromatic number Subtle tree property incompactness
Tag Archives: P-Ideal Dichotomy
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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The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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The P-Ideal Dichotomy and the Souslin Hypothesis
John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading
Dushnik-Miller for regular cardinals (part 3)
Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide … Continue reading