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### 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

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

# 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