= Solution
Assume no induced graph on $P_i$ has the random-graph extension property. For each $i$, choose finite disjoint $A_i,B_i\subseteq P_i$ such that no vertex of $P_i\setminus(A_i\cup B_i)$ realizes the prescribed adjacency pattern. The unions $A=\bigcup_iA_i$ and $B=\bigcup_iB_i$ are finite and disjoint. The extension property in $M$ supplies a new vertex $v$ adjacent to all of $A$ and none of $B$. But $v\in P_j$ for some $j$, contradicting the choice of $A_j,B_j$. Thus some $P_j$ satisfies the extension property. It is a countable model of the <theory of the random graph>, so part (a) gives
$$
\boxed{M\cong P_j\text{ for some }j.}
$$
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