This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127).

**Abstract:** We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization.

A collection of open problems is included.

**Table of Contents:**

- Diamond
- Weak Diamond and the Uniformization Property
- The Souslin Hypothesis and Club Guessing
- Saturation of the Nonstationary Ideal
- Index
- References

**Downloads:**

**Citation information:**

A. Rinot, *Jensen’s diamond principle and its relatives*, Set Theory and Its Applications, Contemp. Math., 533: 125-156, *Amer. Math. Soc., Providence, RI,* 2011.

Update: David Aspero announced a negative answer to Question 13 from our list. The title of his paper is “The consistency of a club-guessing failure at the successor of a regular cardinal”.

Update: Aspero’s paper is in here.

0 likes

In her Oberwolfach talk, Heike Mildenberger just announced a negative answer to Question 17! This is joint work with Shelah (the paper # is 988).

Update: The proof contained a gap, so that the problem remains open.

1 likes

Looks like I solved Question 9. Gotta have some rest; going to sleep…

2 likes

May I know what is the answer?

2 likes

the answer is “yes”. In particular, if GCH holds, and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in $L$.

(though, I am still proofreading).

2 likes

Specifically, we get that GCH+$\square(\aleph_2)$ entails the existence of a $\sigma$-complete $\aleph_2$-Souslin tree.

Update: See here.

2 likes

Pingback: Higher Souslin trees and the GCH, revisited | Assaf Rinot

Update: The answer to Question 16 is “yes”. See here.

0 likes