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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

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

# 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