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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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