### 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
- 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

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

# 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