Understanding the condition of $\diamondsuit^+$

Question

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which satisfy $|\mathcal{A}_\alpha|\le\omega$ for each $\alpha<\omega_1$ and

1. $\mathcal{A}_\alpha\subseteq\mathcal{P}(\alpha)$

2. For each subset $X$ of $\omega_1$ there is a club $C$ such that

   a) $C\subseteq \{\alpha<\omega_1 : X\cap\alpha\in\mathcal{A}_\alpha\}$ and

   b) $C\subseteq \{\alpha<\omega_1 : C\cap\alpha\in\mathcal{A}_\alpha\}$.

(It is just a restatement of $\diamondsuit^+$ given from the Wikipedia.)

I understand the condition 1 and 2a), and they are just a variant of the statements in $\diamondsuit$. However I do not understand the condition 2b). Why $\diamondsuit^+$ contains the condition 2b)? What is the meaning of that? Thanks for any advice!

Answer 1

I can’t give you an intuitive explanation, because I don’t have a good feel for it myself, but I can at least give you an example in which it plays a crucial rôle: $\lozenge^+$ implies the existence of a Kurepa tree, while it’s consistent with $\mathsf{GCH}+\lozenge$ that there be no Kurepa trees. Actually, I’ll show that $\lozenge^+$ implies the existence of a Kurepa family, from which it’s not hard to construct a Kurepa tree.

A Kurepa family is a family $\mathscr{F}\subseteq\wp(\omega_1)$ such that $|\mathscr{F}|\ge\omega_2$, but $\{F\cap\alpha:F\in\mathscr{F}\}$ is countable for each $\alpha<\omega_1$.

Let $\langle\mathscr{A}_\alpha:\alpha<\omega_1\rangle$ be a $\lozenge^+$-sequence. For each $S\in[\omega_1]^{\omega_1}$ and closed $C\subseteq\omega_1$ let

$$F(S,C)=\left\{\alpha\in S:\big[\sigma_C(\alpha),\alpha\big)\cap S=\varnothing\right\}\;,$$

where $\sigma_C(\alpha)$ is the largest element of $C\cup\{0\}$ less than or equal to $\alpha$. (This makes sense, since $C\cup\{0\}$ is closed.) Let $\mathscr{F}$ be the set of all $F(S,C)$ such that $|C|=\omega_1$, and $S\cap\alpha\in\mathscr{A}_\alpha$ and $C\cap\alpha\in\mathscr{A}_\alpha$ for each $\alpha\in C$.

Note that $\lozenge^+$ ensures that for each $S\in[\omega_1]^{\omega_1}$ there is a club $C_S$ such that $F(S,C_S)\in\mathscr{F}$. Now $\lozenge^+$ implies $\lozenge$, which implies $\mathsf{CH}$, so the complete binary tree of height $\omega_1$ has $\omega_1$ nodes and $2^{\omega_1}>\omega_1$ branches. Identify the nodes of the tree with the ordinals less than $\omega_1$; the family of branches is then a family of more than $\omega_1$ uncountable subsets of $\omega_1$, any two of which have countable intersection. Let $S$ and $T$ be two of these sets; there is an $\eta\in C_S\cap C_T$ such that $S\cap T\subseteq\eta$. If $\alpha=\min(S\setminus\eta)$, then $\alpha\in F(S,C_S)\setminus\eta\subseteq S\setminus\eta$, so $\alpha\notin T\supseteq F(T,C_T)$, and therefore $F(S,C_S)\ne F(T,C_T)$. Thus, $|\mathscr{F}|>\omega_1$.

Suppose that $F(S,C)\in\mathscr{F}$ and $\eta<\omega_1$, and let $\alpha=\sigma_C(\eta)$. Suppose first that $\alpha>0$. Then $\alpha\in C$, so $A_S=S\cap\alpha\in\mathscr{A}_\alpha$ and $A_C=C\cap\alpha\in\mathscr{A}_\alpha$. Let $\xi=\min(S\setminus\alpha)$. If $\xi\ge\eta$, then

$$F(S,C)\cap\eta=F(S,C)\cap\alpha=F(A_S,A_C)\;,$$

and if $\xi<\eta$, then

$$F(S,C)\cap\eta=\big(F(S,C)\cap\alpha\big)\cup\{\xi\}=F(A_S,A_C)\cup\{\xi\}\;.$$

Since $\mathscr{A}_\alpha$ and $S\setminus\alpha$ are countable, there are only countably many different possibilities for $F(S,C)\cap\eta$ when $\alpha>0$.

Now suppose that $\alpha=0$. Then $\sigma_C(\beta)=0$ for all $\beta\le\eta$, so $\beta\in F(S,C)\cap\eta$ iff $\beta<\eta$ and $[0,\beta)\cap S=\varnothing$. Thus, either $S\cap\eta=\varnothing$, in which case $F(S,C)\cap\eta=\varnothing$, or $0<\min S<\eta$, and $F(S,C)\cap\eta=\{\min S\}$. This is again only countably many different possibilities for $F(S,C)\cap\eta$.

In short, $\{F\cap\eta:F\in\mathscr{F}\}$ is countable for each $\eta<\omega_1$, and $\mathscr{F}$ is a Kurepa family.