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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

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

# 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