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Lipschitz reduction Hedetniemi's conjecture AIM forcing Antichain Non-saturation Was Ulam right? HOD Well-behaved magma b-scale Strongly Luzin set Knaster Local Club Condensation. Strong coloring GMA Prevalent singular cardinals Almost countably chromatic Chang's conjecture Subadditive Reduced Power approachability ideal specializable Souslin tree C-sequence Minimal Walks perfectly normal Aronszajn tree Nonspecial tree O-space Chromatic number weak square Ostaszewski square positive partition relation weak Kurepa tree Axiom R PFA Forcing with side conditions Large Cardinals Fast club Reflecting stationary set Diamond-sharp stick transformations stationary reflection Analytic sets Cardinal Invariants countably metacompact free Souslin tree Open Access club_AD PFA(S)[S] full tree Subtle cardinal Sakurai's Bell inequality strongly bounded groups indecomposable filter Kurepa Hypothesis Parameterized proxy principle Fat stationary set Small forcing Universal Sequences Selective Ultrafilter Luzin set OCA Intersection model Shelah's Strong Hypothesis Subnormal ideal Iterated forcing Dowker space Slim tree Prikry-type forcing SNR Knaster and friends ccc coloring number L-space Countryman line Forcing Rainbow sets Ineffable cardinal Diamond for trees Forcing Axioms Rock n' Roll Foundations ZFC construction Generalized Clubs super-Souslin tree xbox Martin's Axiom Partition relations for trees Hereditarily Lindelöf space 54G20 Whitehead Problem Ascent Path Uniformization polarized partition relation Erdos Cardinal Precaliber Successor of Singular Cardinal Club Guessing stationary hitting Partition Relations Singular cofinality Distributive tree Singular Density Subtle tree property Almost-disjoint family Monotonically far projective Boolean algebra Successor of Regular Cardinal Square-Brackets Partition Relations Poset Cardinal function S-Space Interval topology on trees Strongly compact cardinal weak diamond higher Baire space diamond star Ascending path Microscopic Approach Closed coloring Commutative cancellative semigroups Uniformly homogeneous Commutative projection system Fodor-type reflection Vanishing levels regressive Souslin tree Sierpinski's onto mapping principle unbounded function very good scale Diamond Cohen real Greatly Mahlo sap square principles Ulam matrix Rado's conjecture Singular cardinals combinatorics Hindman's Theorem Mandelbrot set Postprocessing function Filter reflection Uniformly coherent Jonsson cardinal Absoluteness Generalized descriptive set theory Weakly compact cardinal tensor product graph reflection principles Ramsey theory over partitions Entangled linear order free Boolean algebra incompactness middle diamond Coherent tree square nonmeager set Almost Souslin Sigma-Prikry Souslin Tree Dushnik-Miller Respecting tree P-Ideal Dichotomy Erdos-Hajnal graphs Constructible Universe Amenable C-sequence
Category Archives: Expository
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
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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
A large cardinal in the constructible universe
In this post, we shall provide a proof of Silver’s theorem that the Erdos caridnal $\kappa(\omega)$ relativizes to Godel’s constructible universe. First, recall some definitions. Given a function $f:[\kappa]^{<\omega}\rightarrow \mu$, we say that $I\subseteq\kappa$ is a set of indiscernibles for … Continue reading
c.c.c. forcing without combinatorics
In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing $\mathbb P$ is said to satisfy the c.c.c. iff … Continue reading
