### 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

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

# Tag Archives: Partition Relations

## 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 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