### 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

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

# 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