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### 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

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

# 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