### 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
- 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
- Partitioning the club guessing January 22, 2014

### Keywords

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

# 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

Posted in Blog, Expository
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

## 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