### Archives

### 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

Ascent Path Fat stationary set stationary reflection HOD Microscopic Approach Distributive tree reflection principles Parameterized proxy principle Erdos Cardinal Small forcing square principles PFA Selective Ultrafilter L-space approachability ideal Rainbow sets Rock n' Roll Prevalent singular cardinals Generalized Clubs tensor product graph Almost-disjoint famiy Whitehead Problem very good scale Coherent tree Cohen real diamond star Almost countably chromatic Absoluteness Prikry-type forcing Cardinal Invariants Postprocessing function Successor of Regular Cardinal Kurepa Hypothesis Fast club Nonspecial tree Constructible Universe Souslin Tree Singular cardinals combinatorics Fodor-type reflection 05D10 Shelah's Strong Hypothesis 05A17 11P99 Singular Density coloring number b-scale Hedetniemi's conjecture projective Boolean algebra Successor of Singular Cardinal Chromatic number Rado's conjecture Cardinal function Ostaszewski square S-Space Diamond Uniformly coherent Partition Relations Non-saturation Forcing Axioms free Boolean algebra Dushnik-Miller weak square Square-Brackets Partition Relations Singular coﬁnality Antichain Universal Sequences Weakly compact cardinal Minimal Walks Poset Large Cardinals Foundations Almost Souslin Reduced Power middle diamond incompactness Commutative cancellative semigroups P-Ideal Dichotomy Slim tree xbox Aronszajn tree 20M14 PFA(S)[S] polarized partition relation Axiom R Forcing weak diamond stationary hitting Knaster ccc Erdos-Hajnal graphs Club Guessing Stevo Todorcevic sap OCA Uniformization Hereditarily Lindelöf space square Hindman's Theorem Jonsson cardinal Martin's Axiom Sakurai's Bell inequality Chang's conjecture Mandelbrot set

# 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