### Archives

### Recent blog posts

- Prikry forcing may 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

Almost Souslin sap Kurepa Hypothesis weak diamond Hereditarily Lindelöf space Minimal Walks 05A17 Rado's conjecture Successor of Regular Cardinal Foundations stationary hitting Ostaszewski square Reduced Power Aronszajn tree S-Space Prevalent singular cardinals Sakurai's Bell inequality free Boolean algebra Large Cardinals Slim tree Forcing Axioms Constructible Universe Souslin Tree Generalized Clubs Mandelbrot set Poset PFA Weakly compact cardinal P-Ideal Dichotomy diamond star Prikry-type forcing ccc incompactness stationary reflection very good scale Axiom R Hedetniemi's conjecture Chang's conjecture approachability ideal Erdos-Hajnal graphs Universal Sequences Parameterized proxy principle Rock n' Roll 05D10 Knaster Singular Cofinality Coherent tree Microscopic Approach reflection principles Partition Relations Jonsson cardinal Non-saturation 11P99 Fast club Ascent Path middle diamond Stevo Todorcevic Singular Density Singular coﬁnality Selective Ultrafilter Successor of Singular Cardinal Almost countably chromatic Singular cardinals combinatorics Absoluteness weak square OCA Antichain Diamond b-scale L-space HOD Martin's Axiom Whitehead Problem Commutative cancellative semigroups Uniformization Cardinal function Cohen real Chromatic number xbox Small forcing projective Boolean algebra Almost-disjoint famiy Forcing Fat stationary set Square-Brackets Partition Relations 20M14 Shelah's Strong Hypothesis polarized partition relation Dushnik-Miller square Cardinal Invariants tensor product graph Erdos Cardinal Club Guessing PFA(S)[S] Rainbow sets Hindman's Theorem

# Category Archives: Blog

## 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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## 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