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Prikry-type forcing square principles Ineffable cardinal Uniformly homogeneous Small forcing Selective Ultrafilter diamond star Entangled linear order Interval topology on trees Almost-disjoint family Diamond-sharp Large Cardinals Analytic sets indecomposable filter Forcing Axioms Chromatic number Luzin set Fast club C-sequence regressive Souslin tree Foundations weak square higher Baire space Strongly Luzin set Club Guessing Square-Brackets Partition Relations Sigma-Prikry Subtle cardinal Cohen real Rock n' Roll Forcing with side conditions Local Club Condensation. Rainbow sets Generalized descriptive set theory Fodor-type reflection specializable Souslin tree stationary hitting P-Ideal Dichotomy incompactness Coherent tree Shelah's Strong Hypothesis Diamond Dowker space Commutative projection system Erdos Cardinal Distributive tree Erdos-Hajnal graphs AIM forcing Forcing Amenable C-sequence Closed coloring Singular cofinality stationary reflection Cardinal function Jonsson cardinal Respecting tree PFA(S)[S] Subtle tree property polarized partition relation Subadditive OCA Strong coloring club_AD coloring number Dushnik-Miller positive partition relation Iterated forcing reflection principles approachability ideal Antichain Prevalent singular cardinals S-Space Hedetniemi's conjecture weak diamond ccc Filter reflection Postprocessing function Sierpinski's onto mapping principle sap Singular cardinals combinatorics projective Boolean algebra stick nonmeager set square Non-saturation L-space tensor product graph Successor of Regular Cardinal Vanishing levels Generalized Clubs countably metacompact Ulam matrix full tree Sakurai's Bell inequality Intersection model Ramsey theory over partitions SNR PFA Well-behaved magma free Souslin tree Diamond for trees Microscopic Approach Universal Sequences O-space Ostaszewski square Successor of Singular Cardinal Cardinal Invariants Reduced Power Almost Souslin strongly bounded groups Hindman's Theorem Almost countably chromatic Subnormal ideal Poset ZFC construction Open Access Souslin Tree Slim tree Precaliber Uniformly coherent Reflecting stationary set Singular Density Whitehead Problem Greatly Mahlo Fat stationary set xbox Minimal Walks Parameterized proxy principle perfectly normal Partition Relations transformations middle diamond Hereditarily Lindelöf space 54G20 Kurepa Hypothesis Ascent Path Martin's Axiom Constructible Universe Absoluteness GMA Was Ulam right? Strongly compact cardinal Axiom R Partition relations for trees Countryman line Commutative cancellative semigroups Knaster Chang's conjecture Ascending path Mandelbrot set HOD Monotonically far Lipschitz reduction Nonspecial tree unbounded function Knaster and friends Weakly compact cardinal weak Kurepa tree b-scale super-Souslin tree free Boolean algebra Rado's conjecture very good scale Uniformization Aronszajn tree
Tag Archives: Cardinal Invariants
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
1 Comment
Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … 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
4 Comments
Jones’ theorem on the cardinal invariant $\mathfrak p$
This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading
Bell’s theorem on the cardinal invariant $\mathfrak p$
In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, … Continue reading
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Infinite Combinatorial Topology
Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading
Posted in Notes
Tagged b-scale, Cardinal function, Cardinal Invariants, Hereditarily Lindelöf space
8 Comments
