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