### 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

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

# Tag Archives: 03E02

## Strong failures of higher analogs of Hindman’s Theorem

This is a draft of an upcoming joint paper 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 … 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