Se p 19 97 GENERALIZED COHESIVENESS
نویسندگان
چکیده
We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ram-sey's Theorem. An infinite set A of natural numbers is n–cohesive (respectively, n–r–cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2–coloring of the n–element sets of natural numbers. (Thus the 1–cohesive and 1–r–cohesive sets coincide with the cohesive and r–cohesive sets, respectively.) We consider the degrees of unsolvability and arith-metical definability levels of n–cohesive and n–r–cohesive sets. For example, we show that for all n ≥ 2, there exists a ∆ 0 n+1 n–cohesive set. We improve this result for n = 2 by showing that there is a Π 0 2 2–cohesive set. We show that the n–cohesive and n–r–cohesive degrees together form a linear, non–collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n–cohesive degrees as exactly the degrees ≥ 0 (n+1) and show that each n–r–cohesive degree has jump > 0 (n) .
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