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 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 arithmetical definability levels of n--cohesive and n--r--cohesive sets. For example, we show that for all n 2, there exists a \Delta 0 n+1 n--cohesive set. We improve this result for n = 2 by showing that there is a \Pi 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 s..