Generalized cohesiveness

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A o f natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-c oloring 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, respectiv ely) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n greater than or equal to 2, there exists a Delta(n+1)(0) n-cohesive s et. We improve this result for n = 2 by showing that there is a II20 2-cohe sive set. We show that the n-cohesive and n-r-cohesive degrees together for m a linear, non-collapsing hierarchy of degrees for n 2 2. In addition, for n greater than or equal to 2 we characterize the jumps of n-cohesive degre es as exactly the degrees greater than or equal to 0((n+1)) and also charac terize the jumps of the n-r-cohesive degrees.