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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

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

# 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