Archives
Keywords
Square-Brackets Partition Relations Filter reflection strongly bounded groups Foundations Cohen real Universal Sequences Cardinal function Almost countably chromatic S-Space Respecting tree Uniformly homogeneous Sierpinski's onto mapping principle C-sequence Forcing Strongly Luzin set Singular Density Analytic sets Local Club Condensation. Successor of Singular Cardinal polarized partition relation ZFC construction Ulam matrix regressive Souslin tree Absoluteness Forcing with side conditions Hereditarily Lindelöf space positive partition relation Successor of Regular Cardinal 54G20 Chromatic number Sakurai's Bell inequality Commutative projection system Whitehead Problem Hindman's Theorem Postprocessing function super-Souslin tree Subnormal ideal Closed coloring Dowker space Interval topology on trees Rainbow sets square GMA Knaster and friends Partition relations for trees Was Ulam right? Axiom R Small forcing SNR PFA Forcing Axioms weak diamond full tree Fast club Shelah's Strong Hypothesis Knaster stick Minimal Walks Fodor-type reflection Ascent Path Nonspecial tree approachability ideal Parameterized proxy principle Uniformly coherent Countryman line Sigma-Prikry higher Baire space Ramsey theory over partitions Mandelbrot set diamond star free Souslin tree Antichain Rock n' Roll Slim tree Constructible Universe Ineffable cardinal Hedetniemi's conjecture weak square very good scale PFA(S)[S] Aronszajn tree Open Access AIM forcing unbounded function Coherent tree Erdos-Hajnal graphs Erdos Cardinal Intersection model specializable Souslin tree Souslin Tree stationary reflection Precaliber Fat stationary set Prevalent singular cardinals Non-saturation club_AD coloring number Uniformization Lipschitz reduction free Boolean algebra Ascending path Subtle tree property Poset xbox countably metacompact Iterated forcing Kurepa Hypothesis Amenable C-sequence Diamond reflection principles Jonsson cardinal Strong coloring middle diamond Large Cardinals transformations projective Boolean algebra Rado's conjecture Commutative cancellative semigroups Generalized Clubs Vanishing levels Subadditive Reduced Power Well-behaved magma perfectly normal Almost Souslin Greatly Mahlo Diamond for trees b-scale Singular cofinality indecomposable filter ccc Ostaszewski square incompactness sap nonmeager set Generalized descriptive set theory tensor product graph Entangled linear order Weakly compact cardinal OCA O-space Cardinal Invariants Singular cardinals combinatorics L-space HOD Martin's Axiom Chang's conjecture Prikry-type forcing Almost-disjoint family Reflecting stationary set weak Kurepa tree Club Guessing Microscopic Approach stationary hitting square principles P-Ideal Dichotomy Selective Ultrafilter Subtle cardinal Dushnik-Miller Distributive tree Luzin set Monotonically far Partition Relations Strongly compact cardinal Diamond-sharp
Category Archives: Expository
Prikry Forcing
Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is … Continue reading
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
Leave a comment
The uniformization property for $\aleph_2$
Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a … Continue reading
The Engelking-Karlowicz theorem, and a useful corollary
Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in[\mu]^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading
Kurepa trees and ineffable cardinals
Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. … Continue reading
Variations on diamond
Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little … Continue reading
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
The chromatic numbers of the Erdos-Hajnal graphs
Recall that a coloring $c:G\rightarrow\kappa$ of an (undirected) graph $(G,E)$ is said to be chromatic if $c(v_1)\neq c(v_2)$ whenever $\{v_1,v_2\}\in E$. Then, the chromatic number of a graph $(G,E)$ is the least cardinal $\kappa$ for which there exists a chromatic … Continue reading
Posted in Blog, Expository
Tagged Chromatic number, Erdos-Hajnal graphs, Rado's conjecture, reflection principles
14 Comments
Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … 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
