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

# 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