Archives
Keywords
Sakurai's Bell inequality Almost countably chromatic Square-Brackets Partition Relations Uniformly coherent Respecting tree Luzin set Partition Relations Was Ulam right? Mandelbrot set Jonsson cardinal weak square Cardinal function Chang's conjecture Almost-disjoint family Cardinal Invariants Shelah's Strong Hypothesis S-Space Aronszajn tree Uniformization Forcing Axioms Ramsey theory over partitions incompactness countably metacompact Absoluteness Generalized Clubs PFA(S)[S] club_AD Dushnik-Miller Ascending path Cohen real Countryman line Universal Sequences Club Guessing Entangled linear order Nonspecial tree Small forcing weak diamond Sigma-Prikry xbox specializable Souslin tree free Boolean algebra perfectly normal Diamond-sharp reflection principles Prevalent singular cardinals O-space Ostaszewski square middle diamond Ulam matrix Distributive tree Prikry-type forcing Parameterized proxy principle Partition relations for trees Successor of Regular Cardinal Large Cardinals Knaster and friends Rado's conjecture HOD Strong coloring Subtle tree property Souslin Tree Local Club Condensation. projective Boolean algebra Fast club sap full tree Selective Ultrafilter Commutative cancellative semigroups Fodor-type reflection approachability ideal regressive Souslin tree Knaster strongly bounded groups Vanishing levels Ineffable cardinal Analytic sets stationary reflection unbounded function Amenable C-sequence higher Baire space weak Kurepa tree Lipschitz reduction Postprocessing function Commutative projection system Antichain PFA b-scale square Reduced Power Non-saturation polarized partition relation Hereditarily Lindelöf space Filter reflection C-sequence positive partition relation Forcing with side conditions coloring number tensor product graph Slim tree 54G20 Hedetniemi's conjecture Forcing Constructible Universe Monotonically far Subtle cardinal Foundations Whitehead Problem Well-behaved magma square principles super-Souslin tree stick Axiom R Diamond Poset Successor of Singular Cardinal L-space Uniformly homogeneous very good scale Microscopic Approach indecomposable filter nonmeager set Strongly compact cardinal Rainbow sets SNR Erdos Cardinal Hindman's Theorem Martin's Axiom Closed coloring Greatly Mahlo Chromatic number Dowker space Intersection model stationary hitting Almost Souslin Open Access diamond star Minimal Walks Precaliber Generalized descriptive set theory ccc Rock n' Roll Subadditive transformations Strongly Luzin set GMA Ascent Path free Souslin tree P-Ideal Dichotomy Iterated forcing Interval topology on trees Erdos-Hajnal graphs Sierpinski's onto mapping principle Kurepa Hypothesis Singular Density Diamond for trees Reflecting stationary set Fat stationary set AIM forcing Weakly compact cardinal Singular cardinals combinatorics Coherent tree Singular cofinality Subnormal ideal ZFC construction OCA
Category Archives: Blog
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
Afghan Whigs on Jimmy Fallon
Performing “I’m Her Slave” (from their album Congregation) at NBC’s studios, 22-May-2012:
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
