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