Let $D$ be a Ramsey ultrafilter on $\omega$, and let $F: [\omega]^{n+1} \to \{1,\ldots, k\}$ be a function. Define the function $F_a: [\omega\setminus \{a \}]^n \to \{1, \ldots, k\}$ for each $a\in \omega$, given by $F_a (x) = F( x \cup \{a \})$. Let's suppose that for each $a$, we can find some $H_a \in D$ such that $F_a:[H_a]^{n}\to \{1,\ldots, k\}$ is constant.
Now I'm having some difficulty proving the following statement: "There exists $X \in D$ such that the constant value of $F_a$ is the same for all $a \in X$." This is from Jech(9.2).
Does anyone have any advice about how to prove this?
Thanks very much.
$\bf{NOTE}$: $[X]^n$ means subsets of $X$ of size $n$.