### 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

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

# 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