### Archives

### 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

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

# Tag Archives: Square-Brackets Partition Relations

## 6th European Set Theory Conference, July 2017

I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading

Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
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## Strong failures of higher analogs of Hindman’s Theorem

Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading

## 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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## Complicated Colorings

Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading

Posted in Partition Relations, Publications
Tagged Minimal Walks, Square-Brackets Partition Relations
2 Comments

## MFO workshop in Set Theory, January 2014

I gave an invited talk at the Set Theory workshop in Obwerwolfach, January 2014. Talk Title: Complicated Colorings. Abstract: If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^{\lambda}_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. Downloads:

## Rectangular square-bracket operation for successor of regular cardinals

Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … 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

## Dushnik-Miller for regular cardinals (part 2)

In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading

Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
5 Comments

## CMS Winter Meeting, December 2011

I gave an invited special session talk at the 2011 meeting of the Canadian Mathematical Society. Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals. Abstract: We shall discuss the results of the following papers: Transforming … Continue reading

## Transforming rectangles into squares, with applications to strong colorings

Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading