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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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