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

# 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