Combinatorial Property Vs Computational Property

A set C can be strongly coded under condition < B, C(A) >, where B and C are classes of sets possibly with other parameters, iff there exists A ∈ B such that every Z ∈ C(A) can be used to compute C. The issue is widely studied especially in effective mathematics and reverse mathematics. In this paper, we focus on three kinds of conditions, namely, density condition, enumeration condition and partition condition. For density condition and enumeration condition, we give necessary and sufficient conditions for the parameters that ensure, under the corresponding coding condition < B, C(A) > any set can be strongly computed. As a corollary, we show that for any given C >T 0, if we restrict A to have at least constant density on each member of a computable array of mutually disjoint finite sets then there exists an infinite subset of A that can not be used to compute C. This is in contrast with a well-known result that if A is allowed to have density that approaches to 0, then for any C there exists A such that C can be computed by any infinite G ⊆ A. In addition we give a simplified proof of a main theorem in Greenberg and Miller [5] using a combinatorial result used in the proof of above theorem. As to enumeration condition we also give necessary and sufficient condition for a degree that can be strongly coded under corresponding condition. The last condition we study is partition condition. We give applications of our results including RT2 does not imply WWKL0.