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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

### Keywords

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

# 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