Archives
Keywords
Mandelbrot set weak diamond indecomposable filter Sigma-Prikry specializable Souslin tree Hindman's Theorem Singular cardinals combinatorics Diamond Precaliber nonmeager set Ascent Path free Boolean algebra Rock n' Roll Hedetniemi's conjecture Vanishing levels xbox Uniformly homogeneous countably metacompact middle diamond L-space weak square Strong coloring incompactness strongly bounded groups Chang's conjecture Intersection model perfectly normal Commutative projection system Forcing with side conditions Luzin set Well-behaved magma Erdos Cardinal Forcing Axioms polarized partition relation Hereditarily Lindelöf space Knaster Local Club Condensation. square Jonsson cardinal Prikry-type forcing Ostaszewski square Poset Selective Ultrafilter Analytic sets Singular cofinality Subadditive Singular Density Monotonically far ccc Knaster and friends Antichain club_AD projective Boolean algebra stationary hitting positive partition relation free Souslin tree Lipschitz reduction Cardinal function unbounded function Almost countably chromatic OCA Iterated forcing Interval topology on trees Dowker space Open Access Fodor-type reflection stationary reflection Closed coloring Kurepa Hypothesis Cohen real Uniformly coherent stick AIM forcing Uniformization Subtle tree property Erdos-Hajnal graphs Almost-disjoint family Successor of Regular Cardinal Strongly Luzin set Dushnik-Miller Constructible Universe higher Baire space Reduced Power Weakly compact cardinal Aronszajn tree Filter reflection Ulam matrix Minimal Walks PFA(S)[S] Square-Brackets Partition Relations Shelah's Strong Hypothesis Coherent tree Microscopic Approach Sierpinski's onto mapping principle Generalized descriptive set theory Commutative cancellative semigroups Partition Relations coloring number Postprocessing function Was Ulam right? Subnormal ideal transformations Subtle cardinal Rado's conjecture square principles Ineffable cardinal Forcing ZFC construction Foundations super-Souslin tree full tree Sakurai's Bell inequality Souslin Tree Entangled linear order weak Kurepa tree 54G20 b-scale approachability ideal reflection principles Whitehead Problem O-space tensor product graph Axiom R Fat stationary set Generalized Clubs Large Cardinals Reflecting stationary set Cardinal Invariants P-Ideal Dichotomy Parameterized proxy principle very good scale regressive Souslin tree Prevalent singular cardinals Universal Sequences Diamond-sharp Chromatic number S-Space Countryman line Amenable C-sequence Respecting tree Absoluteness Club Guessing Non-saturation Diamond for trees Fast club HOD Slim tree SNR C-sequence Distributive tree Small forcing Strongly compact cardinal Ascending path Successor of Singular Cardinal Partition relations for trees sap Martin's Axiom Almost Souslin PFA GMA Nonspecial tree Ramsey theory over partitions Greatly Mahlo Rainbow sets diamond star
Category Archives: Blog
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
Review: Is classical set theory compatible with quantum experiments?
Yesterday, I attended a talk at the Quantum Foundations seminar at the beautiful Perimeter Institute for Theoretical Physics (Waterloo, Ontario). The (somewhat provocative) title of the talk was “Is Classical Set Theory Compatible with Quantum Experiments?”, and the speaker was Radu … Continue reading
Comparing rectangles with squares through rainbow sets
In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading
