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Given complete $T$ in countable language, prove for any $kappa >omega$, $exists mathcalMmodels T$ such that $forall AsubseteqmathcalM, mathcalM$ realizes at most $|A|+omega$ types n $S^mathcalM_n(A)$.

Proof goes like this(in Lecture note I'm using). Take $mathcalM$ indiscernible hull of indiscernible $I$ cardinality of $kappa$. Here $mathcalM$ is model of skolemization of $T$ in corresponding language $mathcalL^*$. Given $AsubseteqmathcalM$, $ain A$ can be written as $a=t^mathcalM(c_a)$ where $c_a in I$. take $J=cup_ain A c_a$. then we can show |types over $A$| $leq$|types over J| in $mathcalM$. So it is enough to show latter one$leq |A|+omega$. Now define equivalence relation as follows:

$forall_n<omegaforallbarc,bardin[I]^n barc simbardLeftrightarrow forall_1leq kleq nforall ein J, (c_k<eleftrightarrow d_k<e) wedge (c_k=eleftrightarrow d_k=e)$

where $barc=(c_1,...,c_n),bard=(d_1,...,d_n)$ increasing sequence in $[I]^n$. I'm proving for any term $t(barx)inmathcalL^*$ having n-variables, $barcsimbard$ implies $tp_mathcalM(t^M(barc)/J)=tp_mathcalM(t^M(bard)/J)$. In my note, it is written that "$tp_mathcalM(barc/J)=tp_mathcalM(bard/J)$ implies equality above". but I could prove the former one but I don't know how to get the latter one from first one..

This is an almost immediate consequence of indiscernibility. The intuitive meaning of $overlinecsimoverlined$ is that $overlinec$ and $overlined$ "come in the same order" in $I$ relative to the set $J$. In particular, if $overlinee$ is a finite tuple from $J$, then $overlinecoverlinee$ and $overlinedoverlinee$ are both finite tuples from $I$ which "come in the same order", so they satisfy the same formulas by indiscernibility.

More precisely:

Suppose $varphi(x,overlinee)in texttp_mathcalM(t^M(overlinec)/J)$ and $overlinecsim overlined$. Consider the formula $psi(overlinez,overlinew)$ given by $varphi(t(overlinez),overlinew)$.

We have $mathcalMmodels varphi(t^M(overlinec),overlinee)$, so $mathcalMmodels psi(overlinec,overlinee)$. And by indiscernibility, since $overlinecsimoverlined$, we also have $mathcalMmodels psi(overlined,overlinee)$. So $mathcalMmodels varphi(t(overlined),overlinee)$, and $varphi(x,overlinee)in texttp_mathcalM(t^M(overlined)/J)$. Thus $texttp_mathcalM(t^M(overlinec)/J) = texttp_mathcalM(t^M(overlined)/J)$.