Minimal Pair Constructions and Iterated Trees of Strategies Dedicated to Anil Nerode on the occasion of his sixtieth birthday

0. Introduction. We use the iterated trees of strategies approach developed in [LL1], [LL2] to prove some theorems about minimal pairs. In Sections 1-3, we show how to use these methods to prove the Minimal Pair Theorem of Lachlan [L] and Yates [Y]: Theorem 3.4 (Minimal Pair): There exist nonrecursive r.e. degrees a and b such that a∧b = 0. In Section 4, we add requirements to prove: Theorem 4.3: There are r.e. degrees a and b such that a',b' > 0' and a∧b = 0. And in Section 5, we construct a minimal pair of r.e. degrees a and b such that a' and b' form a minimal pair over 0', i.e., Theorem 5.4: There are r.e. degrees a and b such that a'|b', a∧b = 0, and a'∧b' = 0 ' .