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### 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

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

# 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