### 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

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

# 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