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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

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

# 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