### 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

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

# Tag Archives: polarized partition relation

## Jones’ theorem on the cardinal invariant $\mathfrak p$

This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading