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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

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

# 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