### 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

Forcing Partition Relations Small forcing L-space Cardinal function very good scale approachability ideal Non-saturation sap Universal Sequences OCA free Boolean algebra Foundations Diamond stationary reflection Constructible Universe Dushnik-Miller Cardinal Invariants Coherent tree Antichain Weakly compact cardinal Erdos Cardinal Prevalent singular cardinals Ascent Path Jonsson cardinal Absoluteness Almost countably chromatic Rado's conjecture Singular Cofinality Whitehead Problem Knaster PFA Chromatic number S-Space Almost-disjoint famiy Selective Ultrafilter Mandelbrot set stationary hitting Slim tree Reduced Power 11P99 Poset Axiom R Singular coﬁnality Erdos-Hajnal graphs ccc Hindman's Theorem Singular cardinals combinatorics reflection principles middle diamond Sakurai's Bell inequality 20M14 Cohen real 05D10 Prikry-type forcing weak square Uniformization Ostaszewski square Shelah's Strong Hypothesis weak diamond Martin's Axiom Aronszajn tree Rainbow sets square Rock n' Roll projective Boolean algebra Minimal Walks Successor of Regular Cardinal Microscopic Approach coloring number 05A17 Generalized Clubs incompactness polarized partition relation Fodor-type reflection Fat stationary set Hereditarily Lindelöf space Square-Brackets Partition Relations Kurepa Hypothesis diamond star Club Guessing HOD xbox tensor product graph Successor of Singular Cardinal Almost Souslin b-scale Large Cardinals Forcing Axioms P-Ideal Dichotomy Fast club Commutative cancellative semigroups Chang's conjecture Stevo Todorcevic Hedetniemi's conjecture Parameterized proxy principle Souslin Tree PFA(S)[S] Singular Density

# Category Archives: Partition Relations

## 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

## 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

## 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

## 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