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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

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

# 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