### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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