### 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

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

# Tag Archives: 03E45

## Square with built-in diamond-plus

Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, square, xbox
1 Comment

## Putting a diamond inside the square

Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
1 Comment

## On the consistency strength of the Milner-Sauer conjecture

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading