This is the second part of exercise 3.3.14 of the book "Finite Model Theory by Ebinghaus and Flum"
Let $K$ be a class of finite structures. For any global $n$-ary relation $F$ on $K$ with $n\leq s$.
Assuming than whenever we have : $A, B\in K$ and$\overline{a}\in \Gamma (A), \overline{b}\in B^n$ and the duplicator wins the pebble game $G^s_{\infty}(A,\overline{a}*...*,B,\overline{b}*...*)$ then $\overline{b}\in \Gamma (B)$.
I need to show:
$\Gamma$ is $L^s_{\infty,\omega}$ definable.
As the duplicator has a winning strategy there is a partial $s$-isomorphism $f$ from $A$ to $B$ taking $\overline{a}$ to $\overline{b}$ and as $n\leq s$ there is a partial $n$-isomorphism taking the subsequences of$\overline{a}$ and $\overline{b}$ which are of lenght $n$. Now for each $\overline{a}\in \Gamma (A)$, $f(\overline{a})\in \Gamma (b)$.
How can I get the definability... I think I have to find a formula $\phi$, but how?