### 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

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

# Category Archives: Open Problems

## 6th European Set Theory Conference, July 2017

I gave a 3-lectures tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading

Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
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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

## 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

## Syndetic colorings with applications to S and L

Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading

## The S-space problem, and the cardinal invariant $\mathfrak p$

Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading

Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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## Shelah’s approachability ideal (part 2)

In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading

Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
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## An inconsistent form of club guessing

In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading

## Dushnik-Miller for regular cardinals (part 1)

This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:[\lambda]^2\rightarrow\{0,1\}$ there exists either a subset $A\subseteq \lambda$ of order-type $\lambda$ with … Continue reading

## The order-type of clubs in a square sequence

Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, … Continue reading

## Jensen’s diamond principle and its relatives

This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading