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### 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

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

# 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