I am trying to solve Exercise 6.2.2 in Tent & Ziegler's A Course in Model Theory. The exercise is:
Let $\varphi$ be a formula of Morley rank $\alpha<\infty$ and $\psi_0,\psi_1, \psi_2, …$ an infinite sequence of formulas. Assume that there is a number $k$ such that the conjunction of any $k$ of the $\psi_i$ has Morley rank smaller than $\alpha$. Then $\text{MR}(\varphi\wedge\psi_i)<\alpha$ for almost all $i$.
There is a hint at the end of the book, which says:
Let $I$ be the set of all $i$ such that $\text{MR}(\varphi\wedge\psi_i)=\alpha$. The hypothesis implies that all $k$-element subsets of $I$ contain two indices $i,j$ such that $\varphi\wedge\psi_i\not\sim_\alpha\varphi\wedge\psi_j$. So $|I|\leq (k-1)MD(\varphi)$.
I am lost on how the hypothesis implies the statement in the hint and also how that statement implies $|I|\leq (k-1)MD(\varphi)$. It seems like I have been going in circles.
My attempt:
Let $i_1,…,i_k$ be $k$ elements from $I$. Assume for contradiction that $\varphi\wedge\psi_{i_1}\sim_\alpha…\sim_\alpha\varphi\wedge\psi_{i_k}$. By definition, $\text{MR}(\varphi\wedge \psi_{i_m}\triangle \varphi\wedge \psi_{i_n})<\alpha$ for any $m,n\leq k$. Because $\wedge$ distributes over $\triangle$, we have $\text{MR}(\varphi\wedge (\psi_{i_m}\triangle\psi_{i_n}))<\alpha$. At this point, write $\varphi$ as the disjunction witnessing its Morley degree, $\varphi\equiv \varphi_1\vee…\vee\varphi_d$ so $\text{MR}(\varphi\wedge (\psi_{i_m}\triangle\psi_{i_n}))=\text{MR}(\varphi_l\wedge (\psi_{i_m}\triangle\psi_{i_n}))$ for some $\varphi_l$ that is $\alpha$ strongly minimal. Then I tried writing out $(\psi_{i_m}\triangle\psi_{i_n})$, but I think I am moving further and further away from being able to apply the assumption.
Best Answer
Let $d = \mathrm{MD}(\varphi)$, and let $p_1,\dots,p_d$ enumerate the complete types of Morley rank $\alpha$ containing $\varphi$.
For each $1\leq i\leq d$, $p_i$ contains at most $(k-1)$ of the formulas $\psi_n$. Indeed, if $p_i$ contains $\psi_{n_1},\dots,\psi_{n_k}$, then $\bigwedge_{j=1}^k \psi_{n_j}\in p_i$, contradicting the hypothesis that $\mathrm{MR}(\bigwedge_{j=1}^k \psi_{n_j})<\alpha$.
It follows that only finitely many (at most $d(k-1)$) of the formulas $\psi_n$ are contained in any of the types $p_i$ for $1\leq i \leq d$. And if $\psi_n$ is a formula which is not contained in any of the types $p_i$, then $\varphi\land \psi_n$ is not contained in any type of Morley rank $\geq \alpha$, so $\mathrm{MR}(\varphi\land \psi_n)<\alpha$.