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### 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

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

# 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