Let k be an infinite cardinal. A subset of $(^k k)^n $ is a $\Sigma _1^1 $ -subset if it is the projection p[T] of all cofinal branches through a subtree T of $(lt;kk)^{n + 1} $ of height k. We define $\Sigma _k^1 - ,\Pi _k^1 $ - and $\Delta _k^1$ subsets of $(^k k)^n $ as usual. Given an uncountable regular cardinal k with k = k (...) there is a shrink)