Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb P=\left\{(p,A)\mid p\in[\kappa]^{<\omega}, A\in\mathcal U, \max p<\min A\right\}.$$
We also define the order, $(q,B)$ is stronger than $(p,A)$ if $q$ is an end-extension of $p$, $B\subseteq A$ and $q\setminus p\subseteq A$. The generic $G$ adds an $\omega$-sequence, $x_G=\bigcup\{q\mid\exists A:(q,A)\in G\}$ which we call a *Prikry sequence*. We can show that $x\subseteq^* A$, for all $A\in\cal U$, that is $x\setminus A$ is finite.

In the other direction, if $M$ is a model of $\sf ZFC$ in which $\kappa$ is measurable, and $M\subseteq V$, such that in $V$ we have some $x\in[\kappa]^\omega$ such that for some $\cal U$ in $M$ which is a normal measure on $\kappa$, $x\subseteq^*A$ for all $A\in\cal U$, then $x=x_G$ for some generic filter $G$ over the Prikry forcing defined from $\cal U$.

One conclusion from this last statement is that if $x$ is a Prirky sequence for $\kappa$, and $y\subseteq x$ is infinite then $y$ is also a Prikry sequence. This raises the following question.

Suppose that $x$ is a Prikry sequence for $\kappa$, and $y\subseteq x$ is infinite. Is $x$ generic over $y$? More generally, if $y,w\subseteq x$ are disjoint infinite subsets, are they pairwise generic? What about the weaker condition $y\cap w$ being finite?