Mutually Indiscernible Sequences

If $$C$$ is a small set of parameters, a set of $$C$$-indiscernible sequences is said to be mutually indiscernible if each sequence in the set is indiscernible over the union of $$C$$ and the other indiscernible sequences. For example, $$a_1, a_2, \ldots$$ and $$b_1, b_2, \ldots$$ are mutually indiscernible if $$a_1, a_2, \ldots$$ is an indiscernible sequence over $$\{b_1, b_2, \ldots\}$$, and $$b_1, b_2, \ldots$$ is an indiscernible sequence over $$a_1, a_2, \ldots$$.

Mutually indiscernible sequences can be stretched, extended, and extracted just like indiscernible sequences. For example, for two sequences, one has the following

Let $$a_1, a_2, \ldots$$ and $$b_1, b_2, \ldots$$ be two sequences, and let $$C$$ be a set of parameters. Then, passing to an elementary extension if necessary, there exist sequences $$a_1', a_2', \ldots$$ and $$b_1', b_2', \ldots$$, mutually indiscernible over $$C$$, that are related to the original sequences in the following way: if a formula $$\phi(a_1',a_2',\ldots,a_n',b_1',b_2',\ldots,b_m',c)$$ holds for some $$n, m$$, and some $$c \in C$$, then there exist $$i_1 < \cdots < i_n$$ and $$j_1 < \cdots < j_m$$ such that $$\phi(a_{i_1},\ldots,a_{i_n},b_{j_1},\ldots,b_{j_n},c)$$ holds.

For example, if the $$a_i$$ lived in a sort with a partial ordering, and if $$a_1 < a_2 < \cdots$$, then $$a_1' < a_2' < \cdots$$ will hold. If $$\phi(a_i;b_j)$$ held for every $$i, j$$, then $$\phi(a'_i;b'_j)$$ will hold for every $$i,j$$.

This theorem doesn't seem to follow directly from Ramsey's theorem, but can be proven using Morley sequences of global invariant types in the same way that Ramsey's theorem can be proven. One uses the following fact: if $$p$$ and $$q$$ are two global $$C$$-invariant types, and $$a_1, a_2, \ldots, b_1, b_2, \ldots$$ realizes $$p^{\otimes \omega} \otimes q^{\otimes \omega} | C$$, then $$a_1, a_2, \ldots$$ and $$b_1, b_2, \ldots$$ will be mutually indiscernible over $$C$$.

(If somebody knows a direct proof using Ramsey's theorem, they should add it here.)

The ability to extract mutually indiscernible sequences is an important tool for working with ict and inp patterns.