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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

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

# 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
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## 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
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## 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