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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

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

# 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