### 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

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

# 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