### 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

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

# Tag Archives: Singular cardinals combinatorics

## More notions of forcing add a Souslin tree

Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading

## Ordinal definable subsets of singular cardinals

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading

Posted in Publications, Singular Cardinals Combinatorics
Tagged HOD, Singular cardinals combinatorics
2 Comments

## Dushnik-Miller for singular cardinals (part 2)

In the first post on this subject, we provided a proof of $\lambda\rightarrow(\lambda,\omega+1)^2$ for every regular uncountable cardinal $\lambda$. In the second post, we provided a proof of $\lambda\rightarrow(\lambda,\omega)^2$ for every singular cardinal $\lambda$, and showed that $\lambda\rightarrow(\lambda,\omega+1)^2$ fails for every … Continue reading

Posted in Blog, Expository
Tagged Dushnik-Miller, Partition Relations, Singular cardinals combinatorics
27 Comments

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

Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Proof. Suppose that $\lambda$ is a singular cardinal, and $c:[\lambda]^2\rightarrow\{0,1\}$ is a given coloring. For any ordinal $\alpha<\lambda$, denote … Continue reading

## On topological spaces of singular density and minimal weight

Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading

## Young Researchers in Set Theory, March 2011

These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … Continue reading

## Workshop on Set Theory and its Applications, February 2007

These are the slides of a talk given at the Workshop on Set Theory and its Applications workshop (Weizmann Institute, February 19, 2007). Talk Title: Nets of spaces having singular density Abstract: The weight of a topological space X is the … Continue reading