### 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

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

# 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