### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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