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Fix an r-coloring of the pairs of natural numbers. An ordered list of distinct integers, is a monochromatic path for color k, if every two adjacent integers in the list have the same color. A path decomposition for the coloring is a collection of r paths P0,P1,...,Pr−1 such that Pj is a path of color j and every integer appears on exactly one path. Improving on an unpublished result of Erdos, Rado published a theorem which implies: Every r- coloring of the pairs of natural numbers has a path decomposition. We analyse the effective content of this theorem. 

The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are all very close to total: joining a continuous degree with a total degree that is not below it always results in a total degree. We call this property almost totality. We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of enumeration degrees, we see that the continuous degrees are also definable. Applying earlier work on the continuous degrees, this shows that the relation “PA above” on the total degrees is definable in the enumeration degrees. In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly. We prove that the enumeration degree of A is continuous if and only if A is codable, meaning that A is enumeration above the complement of an infinite tree, every path of which enumerates A.