### 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

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

# Tag Archives: Dushnik-Miller

## Dushnik-Miller for regular cardinals (part 3)

Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide … Continue reading

## 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 regular cardinals (part 2)

In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading

Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
5 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

## Dushnik-Miller for regular cardinals (part 1)

This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:[\lambda]^2\rightarrow\{0,1\}$ there exists either a subset $A\subseteq \lambda$ of order-type $\lambda$ with … Continue reading