### 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

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

# 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 Preprints, Singular Cardinals Combinatorics
Tagged HOD, Singular cardinals combinatorics
1 Comment

## 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 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

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