Consider Prikry's forcing for changing the cofinality of a measurable cardinal into $\omega.$ The forcing has the Prikry property and one can prove this either directly or using Rowbottom's theorem which is a Ramsey type theorem.


Now consider Magidor's forcing for changing the cofinality of a large cardinal into $\omega\_1$ (or any other uncountable regular cardinal). The proof of Prikry property for it (that I have seen) is direct.


My question is:



> 
> **Question.** Is there a suitable Ramsey type theorem such that one can use it to present a simplified proof of the Prikry property for Magidor's forcing?
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Any references, if there are, are appreciated.



Ramsey type theorems and Magidor's forcing
There is an analog of Rowbottom's theorem which can be extracted from the proof of the Magidor forcing:


Suppose that $u$ is a sequence of ultrafilters on $\kappa$ of length $\lambda<\kappa$ and $f\colon [\kappa]^{<\omega}\to 2$. Then there are sets $A\_\nu\in u(\nu)$ for $\nu<\lambda$ such that whenever $\langle \nu\_0,\dots,\nu\_{k-1}\rangle$ is a finite sequence of ordinals less than $\lambda$, the function $f$ is constant on the set of increasing sequences $\langle \alpha\_0,\dots,\alpha\_{k-1}\rangle\in A\_{\nu\_0}\times\dots\times A\_{\nu\_{k-1}}$. 


This can be extended to Radin forcing on sequences of ultrafilters of length less than $\kappa^+$; I'm not sure what could be done for longer sequences.


