  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 $n \ge 2$, 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 $n \geq 2$. In addition, for $n \geq 2$ we characterize the jumps
of $n$--cohesive degrees as exactly the degrees ${\bf \geq \jump{0}{(n+1)}}$
and show that each $n$--r--cohesive degree has jump ${\bf > \jump{0}{(n)}}$.
