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Let $\kappa$ be a measurable cardinal and let $\mathcal{U}$ be a normal measure on $\kappa$. Let $\mathbb{P}$ be the standard Prikry forcing using $\mathcal{U}$. Let $\mathbb{Q} = \text{Add}(\kappa, 1)$ be Cohen forcing for adding a new subset to $\kappa$ using partial functions from $\kappa$ to $2$ of size ${<}\kappa$.

Question: Is there a projection from $\mathbb{P}$ to $\mathbb{Q}$?

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The following result of Tom Benhamou and Gitik might be related:

Theorem. Suppose $V$ satisfies $GCH$ and $\kappa$ is a measurable cardinal. Then in a cofinality preserving generic extension, there exists a $\kappa$-complete ultrafilter $U$ on $\kappa$ such that Prikry forcing with $U$ adds a Cohen subset of $\kappa$ over $V$.

See page 69 of the paper Sets in Prikry and Magidor Generic Extensions.

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  • $\begingroup$ So the assumption of normality (or at least that the ultrafilter is P-point) is necessary in the theorem of Gitik, Kanovei and Koepke. $\endgroup$
    – Yair Hayut
    Commented Jan 3, 2018 at 8:10
  • $\begingroup$ Note that the Cohen is generic over ground model, so it does not address the above result of Gitik-Kanovei-Koepke $\endgroup$
    – Mohammad Golshani
    Commented Jan 3, 2018 at 12:09

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