### 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

Large Cardinals Rado's conjecture P-Ideal Dichotomy Poset PFA Postprocessing function Nonspecial tree Generalized Clubs Almost Souslin Fat stationary set Axiom R Square-Brackets Partition Relations Ostaszewski square Erdos Cardinal diamond star incompactness Almost-disjoint famiy projective Boolean algebra Fast club Rock n' Roll Partition Relations Ascent Path Weakly compact cardinal stationary hitting super-Souslin tree free Souslin tree Selective Ultrafilter Minimal Walks Aronszajn tree Universal Sequences Stevo Todorcevic OCA Souslin Tree Cohen real Sakurai's Bell inequality tensor product graph Knaster Hindman's Theorem Luzin set xbox Almost countably chromatic square principles Parameterized proxy principle Distributive tree Kurepa Hypothesis Chromatic number Constructible Universe Coherent tree weak diamond approachability ideal Jonsson cardinal Rainbow sets reflection principles HOD Forcing Mandelbrot set Hedetniemi's conjecture sap Singular Density Successor of Singular Cardinal 05A17 middle diamond Slim tree stationary reflection L-space Chang's conjecture Uniformly coherent Martin's Axiom free Boolean algebra Cardinal function coloring number Absoluteness Prevalent singular cardinals weak square square Singular cardinals combinatorics Cardinal Invariants Microscopic Approach polarized partition relation Prikry-type forcing Non-saturation ccc 11P99 Uniformization Forcing Axioms Shelah's Strong Hypothesis Foundations Whitehead Problem Antichain Erdos-Hajnal graphs Fodor-type reflection b-scale PFA(S)[S] Hereditarily Lindelöf space specializable Souslin tree Singular coﬁnality S-Space Small forcing Commutative cancellative semigroups Dushnik-Miller Club Guessing Diamond very good scale Reduced Power Successor of Regular Cardinal

# 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