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

# Tag Archives: Rainbow sets

## Prolific Souslin trees

In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading

Posted in Blog, Expository
Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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## Polychromatic colorings

These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function $f:A\rightarrow B$ is said to … Continue reading

## Comparing rectangles with squares through rainbow sets

In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading