Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of $L$-formulas in $n$ free variables then is it possible to give a coloration $c:[X]^n\longrightarrow 2^{|L|}$ such that:
$$ c(\bar{a})=c(\bar{b}) \Longleftrightarrow \frak{A} \models \varphi[\bar{a}]\longleftrightarrow \varphi[\bar{b}] \tag{i} $$
For all $\varphi \in \Gamma$
How to give a coloration such that indiscernible sets are of homogeneous color?. Is it possible to do so without actually stating $(i)$ in the definition?. Is there any way to do this under some other set of hypothesis maybe?
It's easier to define $c$ to map $[X]^n$ into the set ${}^\Gamma\{\top,\bot\}$ of functions from $\Gamma$ to truth values, rather than into the cardinal $2^{|L|}$, and it doesn't matter for the result since there's a bijection between ${}^\Gamma\{\top,\bot\}$ and a subset of $2^{|L|}$. So let $c(\overline a)$ be the function assigning to each $\phi\in\Gamma$ the truth value of $\phi(\overline a)$ in $\mathfrak A$.