# The projection of a weakly homogeneous tree is determined

Where can I read a proof of this?

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Without the assumption that there is a Woodin cardinal, this can fail. For example, every countable tree is weakly homogeneous via principal measures, but in $L$ there is a countable tree whose projection is not determined. (There are also some less trivial examples.) On the other hand, every homogeneous tree is determined, and a proof of this is implicit in Martin's proof of $\Pi^1_1$ determinacy from sharps as mentioned in Andres's answer below. – Trevor Wilson Jun 17 '13 at 19:23

And probably you want to read first Martin's proof of determinacy of $\Pi^1_1$ from sharps, since the key ideas are there (this should be in Jech's or Kanamori's book).
A modern, very quick and nice exposition of this and the key related ideas in proving determinacy at the projective level is in Itay Neeman's paper "Determinacy in $L({\mathbb R})$", in the Handbook of Set Theory. You can currently download the paper from Itay's page.