### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014

### Keywords

Slim tree Almost-disjoint famiy Rado's conjecture Singular Density Large Cardinals square Ascent Path Rock n' Roll Chromatic number Club Guessing L-space Uniformization Prikry-type forcing Singular coﬁnality Almost Souslin Diamond square principles Forcing b-scale Successor of Singular Cardinal Partition Relations Antichain Microscopic Approach Hedetniemi's conjecture Parameterized proxy principle weak diamond diamond star Distributive tree Non-saturation Souslin Tree specializable Souslin tree incompactness Erdos Cardinal projective Boolean algebra middle diamond Small forcing very good scale Ostaszewski square Uniformly coherent Coherent tree PFA Whitehead Problem Constructible Universe S-Space Singular cardinals combinatorics Knaster Hereditarily Lindelöf space Jonsson cardinal Cohen real Successor of Regular Cardinal Erdos-Hajnal graphs Shelah's Strong Hypothesis polarized partition relation approachability ideal Almost countably chromatic Poset reflection principles Foundations Nonspecial tree Chang's conjecture Square-Brackets Partition Relations stationary hitting Selective Ultrafilter P-Ideal Dichotomy Weakly compact cardinal sap Reduced Power super-Souslin tree Absoluteness Dushnik-Miller Fat stationary set OCA Sakurai's Bell inequality Hindman's Theorem Fodor-type reflection Kurepa Hypothesis free Boolean algebra Rainbow sets free Souslin tree Martin's Axiom Commutative cancellative semigroups stationary reflection Fast club Stevo Todorcevic Postprocessing function coloring number tensor product graph weak square PFA(S)[S] Prevalent singular cardinals Universal Sequences Mandelbrot set Generalized Clubs ccc Cardinal function Cardinal Invariants Axiom R HOD Minimal Walks Luzin set Forcing Axioms Aronszajn tree xbox

# 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

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## 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

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Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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## 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

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Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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## 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