### 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
- Genearlizations 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

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

# 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 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