Archives
Keywords
Axiom R square principles Microscopic Approach Foundations b-scale 54G20 Diamond for trees Aronszajn tree Ascent Path Fodor-type reflection Prevalent singular cardinals projective Boolean algebra Jonsson cardinal Countryman line Filter reflection Chromatic number Shelah's Strong Hypothesis free Boolean algebra coloring number specializable Souslin tree Cardinal function Almost Souslin Vanishing levels sap Absoluteness Generalized Clubs Coherent tree Whitehead Problem Iterated forcing weak square Large Cardinals Distributive tree P-Ideal Dichotomy Club Guessing Small forcing Kurepa Hypothesis Diamond-sharp Subadditive Uniformization Partition Relations Knaster Strongly Luzin set C-sequence Dowker space Chang's conjecture SNR Rado's conjecture Uniformly homogeneous PFA Fast club S-Space Diamond Square-Brackets Partition Relations OCA regressive Souslin tree L-space positive partition relation Rock n' Roll Nonspecial tree Reflecting stationary set Successor of Singular Cardinal club_AD Cohen real Subnormal ideal Commutative projection system Weakly compact cardinal unbounded function Subtle tree property Sigma-Prikry Slim tree Analytic sets PFA(S)[S] Reduced Power Forcing Axioms Erdos-Hajnal graphs weak Kurepa tree indecomposable ultrafilter Precaliber full tree Rainbow sets Ulam matrix Lipschitz reduction middle diamond Dushnik-Miller Sakurai's Bell inequality Hereditarily Lindelöf space Strong coloring transformations Ramsey theory over partitions strongly bounded groups stick stationary reflection Generalized descriptive set theory Greatly Mahlo higher Baire space Hindman's Theorem polarized partition relation Antichain HOD Local Club Condensation. Prikry-type forcing nonmeager set GMA super-Souslin tree Poset Commutative cancellative semigroups approachability ideal Hedetniemi's conjecture Cardinal Invariants Postprocessing function tensor product graph incompactness square stationary hitting Mandelbrot set Singular cardinals combinatorics ccc Selective Ultrafilter Singular cofinality Parameterized proxy principle xbox countably metacompact Forcing Uniformly coherent Constructible Universe O-space Successor of Regular Cardinal diamond star Minimal Walks Well-behaved magma Luzin set very good scale reflection principles Knaster and friends ZFC construction Souslin Tree AIM forcing Singular Density Ostaszewski square Open Access Closed coloring Was Ulam right Sierpinski's onto mapping principle free Souslin tree Martin's Axiom Intersection model Subtle cardinal weak diamond Respecting tree Almost countably chromatic Erdos Cardinal Almost-disjoint family Strongly compact cardinal Non-saturation Fat stationary set Universal Sequences Ineffable cardinal Amenable C-sequence
Category Archives: Expository
Prolific Souslin trees
In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading
Posted in Blog, Expository
Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
Leave a comment
Generalizations of Martin’s Axiom and the well-met condition
Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading
Many diamonds from just one
Recall Jensen’s diamond principle over a stationary subset $S$ of a regular uncountable cardinal $\kappa$: there exists a sequence $\langle A_\alpha\mid \alpha\in S \rangle$ such that $\{\alpha\in S\mid A\cap\alpha=A_\alpha\}$ is stationary for every $A\subseteq\kappa$. Equivalently, there exists a sequence $\langle … Continue reading
Square principles
Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading
Partitioning the club guessing
In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading
Walk on countable ordinals: the characteristics
In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. … Continue reading
Polychromatic colorings
These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function $f:A\rightarrow B$ is said to … Continue reading
Syndetic colorings with applications to S and L
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
Open coloring and the cardinal invariant $\mathfrak b$
Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading
PFA and the tree property at $\aleph_2$
Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, … Continue reading
