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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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