Archives
Keywords
Poset Amenable C-sequence Forcing Axioms Strong coloring Strongly Luzin set Uniformly homogeneous Analytic sets 54G20 Club Guessing HOD free Boolean algebra Chromatic number nonmeager set projective Boolean algebra GMA C-sequence Hereditarily Lindelöf space Successor of Singular Cardinal Fast club Rainbow sets countably metacompact Coherent tree Absoluteness Ineffable cardinal Vanishing levels Singular cofinality Lipschitz reduction Generalized descriptive set theory Mandelbrot set O-space stationary hitting Precaliber Kurepa Hypothesis Small forcing Partition Relations Was Ulam right Selective Ultrafilter Postprocessing function Filter reflection Nonspecial tree sap P-Ideal Dichotomy Distributive tree L-space Minimal Walks regressive Souslin tree unbounded function transformations Almost Souslin AIM forcing PFA very good scale Singular cardinals combinatorics Subtle tree property approachability ideal diamond star PFA(S)[S] stick Ulam matrix Souslin Tree stationary reflection Knaster Erdos-Hajnal graphs super-Souslin tree Rock n' Roll Prikry-type forcing Reflecting stationary set Erdos Cardinal Non-saturation tensor product graph Universal Sequences Successor of Regular Cardinal Subadditive Cardinal Invariants Greatly Mahlo reflection principles Diamond strongly bounded groups SNR Diamond for trees Uniformization Cohen real square principles higher Baire space Sakurai's Bell inequality Sigma-Prikry middle diamond ZFC construction Iterated forcing Shelah's Strong Hypothesis Almost-disjoint family Knaster and friends Weakly compact cardinal coloring number Cardinal function Slim tree free Souslin tree Parameterized proxy principle Antichain Hindman's Theorem Luzin set weak diamond Open Access Reduced Power Prevalent singular cardinals indecomposable ultrafilter Sierpinski's onto mapping principle polarized partition relation Square-Brackets Partition Relations xbox Local Club Condensation. Generalized Clubs specializable Souslin tree Dowker space Fat stationary set Almost countably chromatic Singular Density Ascent Path Ramsey theory over partitions OCA Whitehead Problem Jonsson cardinal Axiom R Constructible Universe Chang's conjecture Large Cardinals Subtle cardinal S-Space positive partition relation weak square Forcing Dushnik-Miller Aronszajn tree Diamond-sharp Rado's conjecture Closed coloring incompactness Fodor-type reflection Uniformly coherent Well-behaved magma club_AD Martin's Axiom ccc square full tree Hedetniemi's conjecture Ostaszewski square Commutative cancellative semigroups Foundations Subnormal ideal Microscopic Approach b-scale
Category Archives: Blog
A strong form of König’s lemma
A student proposed to me the following strong form of König’s lemma: Conjecture. Suppose that $G=(V,E)$ is a countable a graph, and there is a partition of $V$ into countably many pieces $V=\bigcup_{n<\omega}V_n$, such that: for all $n<\omega$, $V_n$ is … Continue reading
Posted in Blog
2 Comments
Prikry forcing may add a Souslin tree
A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading
The reflection principle $R_2$
A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading
Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
Leave a comment
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