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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

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

# 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