### 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

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

# 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