### 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

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

# Tag Archives: Minimal Walks

## 6th European Set Theory Conference, July 2017

I gave a 3-lectures 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
4 Comments

## Distributive Aronszajn trees

Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading

## Square with built-in diamond-plus

Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, square, xbox
1 Comment

## Chain conditions of products, and weakly compact cardinals

Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading

Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments

## 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:

## Walk on countable ordinals: the characteristics

In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. … Continue reading

## 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

## Young Researchers in Set Theory, March 2011

These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … 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