### 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
- Genearlizations 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

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

# 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 is a proper class of uncountable cardinals $\kappa$ … 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