# The Diamond Principle and Whitehead's problem

I am currently writing my master's thesis about Whitehead's problem in homological algebra. My main source is Eklof's article https://www.jstor.org/stable/2318684. Eklof does not name the Diamond principle in the article (though I know Shelah uses it in his original proof), but instead uses the following axiom, which he lists as a consequence of V=L.

Let C be a set which is the union of a strictly increasing smooth chain of countable sets $$\{C_{\upsilon}\ |\ \upsilon<\omega_{1}\}$$ (smoothness means that $$C_\lambda = \bigcup_{\upsilon<\lambda} C_\upsilon$$ for $$\lambda$$ a limit ordinal), and let $$E$$ be a stationary subset of $$\omega_{1}$$. Then there is a sequence $$\{S_{\upsilon}\ |\ \upsilon\in E\}$$ such that $$S_{\upsilon}\subseteq C_{\upsilon}$$ for all $$\upsilon\in E$$ and such that for any subset $$X$$ of $$C$$, the set of $$\upsilon\in E$$ with $$X\cap C_{\upsilon} = S_{\upsilon}$$ is stationary in $$\omega_{1}$$.

On my first read of the article, I assumed this was either equivalent to or a simple consequence of the Diamond Principle. However, after actually trying to prove it, I'm not so sure anymore. I have not been able to construct a proof, and I'm not even sure this is actually the Diamond Principle. Are there any users out there who might shed some light on this? If this axiom is a consequence of $$\diamondsuit$$, how do I prove this? If not, how is it related to $$\diamondsuit$$?

• What does it mean for a chain of sets to be "smooth"? Commented Nov 11, 2021 at 20:01
• Oh, I suppose that's not a common term. Let me update. Commented Nov 11, 2021 at 20:08

To see that $$\lozenge$$ follows from this principle, simply set $$C=\omega_1$$ with $$C_\nu=\nu$$ and take $$E=\rm Lim$$, the club of limit ordinals. Then for every subset of $$C$$, that is $$\omega_1$$, the set of correct guesses is stationary.
In the opposite direction, it seems to me that Eklof defines $$\lozenge_E$$, which in general is stronger than $$\lozenge$$. Of course, under $$V=L$$ none of this matters, and ultimately, this is just a question of what is the most convenient tool for explaining Shelah's proof.
• I was really hoping that $\diamondsuit$ was sufficient, as I was trying to do a comparative analysis of the independence of Whitehead's problem and Suslin's problem. As such, I'd love to use the exact same extensions of ZFC to prove each, and I was hoping to avoid going through all of V = L. Do you have a reference on $\diamondsuit_E$, so I can read a little more about it? Commented Nov 11, 2021 at 22:03
• You can find it mentioned in all the usual places like Jech and such (often written as $\lozenge(E)$. Assaf Rinot has a nice write up about diamond and its relatives. As for the proof that it is stronger, I couldn't track a reference, and had to ask a colleague to make sure I remembered correctly. Commented Nov 12, 2021 at 9:47