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