### 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

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

# 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