### Archives

### 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

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

# 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