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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

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

# 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