### Archives

### Recent blog posts

- More notions of forcing add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations 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
- Partitioning the club guessing January 22, 2014

### Keywords

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

# Category Archives: Blog

## More notions of forcing 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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## Genearlizations 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