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