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

# 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