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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

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

# Tag Archives: 03E02

## 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

## 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