Let $\mathcal{U}$ be an ultrafilter over $\omega$, and let $\mathcal{X} \subseteq [\omega]^\omega$. In two separate texts, there are two possible interpretations of a $\mathcal{U}$-Ramsey set, as described below (Definition 7.37 and Definition 3). My question is:
Do these two definitions coincide?
Interpretation 1. The first interpretation is found in Stevo Todorcevic's book Introduction to Ramsey Spaces.
Definition 7.29. For any ultrafilter $\mathcal{U}$ over $\omega$ (not necessarily Ramsey), we define a $\mathcal{U}$-tree to be a subtree $T$ of ${}^{<\omega}\omega$ (finite subsets of $\omega$, not sequences) with the property that for every finite subset $t \subseteq \omega$ such that $\operatorname{stem}(T) \subseteq t$, we have: $$ \{n \in \omega : t \cup \{n\} \in T\} \in \mathcal{U} $$ Definition 7.30. For two $\mathcal{U}$-trees $T'$ and $T$, we say that $T'$ is a pure refinement of $T$ if $\operatorname{stem}(T') = \operatorname{stem}(T)$ and $T' \subseteq T$.
Definition 7.37. We then say that $\mathcal{X}$ is $\mathcal{U}$-Ramsey if for every $\mathcal{U}$-tree $T$, there is a pure refinement $T'$ of $T$ such that $[T'] \subseteq \mathcal{X}$ or $[T'] \subseteq \mathcal{X}^c$.
Interpretation 2. I first saw this interpretation in the paper Happy and mad families in $L(\mathbb{R})$, but I believe this is quite a standard definition.
Definition 1. If $y_0 \supseteq y_1 \supseteq y_2 \supseteq \cdots$ is a decreasing sequence of subsets of $\omega$, then we call a set $y_\infty \in [\omega]^\omega$ a diagonalisation of the sequence $\langle{y_n : n < \omega}\rangle$ iff $f(n+1) \in y_{f(n)}$ for every $n < \omega$, where $f : \omega \to \omega$ is the increasing enumeration of $y_\infty$.
Definition 2. A $\mathcal{H} \subseteq [\omega]^\omega$ is called a coideal if is satisfies the conditions:
(Upward-closure) If $x \in H$ and $y \supseteq x$, then $y \in H$.
(Pigeonhole) If $x_0 \cup \cdots \cup x_n \in H$, then $x_k \in H$ for some $k$.
Furthermore, $\mathcal{H}$ is said to be selective if it satisfies the condition:
- (Selectivity) Every decreasing sequence $y_0 \supseteq y_1 \supseteq \cdots$ of members of $\mathcal{H}$ has a diagonalisation in $H$.
A selective coideal is also called a happy family.
Definition 3. If $H$ is a coideal (typically a happy family) and $\mathcal{X} \subseteq [\omega]^\omega$ is a set of reals, then we say $\mathcal{X}$ is $\mathcal{H}$-Ramsey if there is $M \in \mathcal{H}$ such that $[M]^\omega \subseteq X$ or $[M]^\omega \subseteq X^c$.
Note that if $\mathcal{H}$ is an ultrafilter, then it is a happy family iff it is a Ramsey ultrafilter.