Ramsey's theorem for computably enumerable colorings

It is shown that for each computably enumerable set P of n-element subsets of m there is an infinite Pi (0)(n) set A subset of or equal to omega such that either all n-element subsets of ii are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Pi (0)(n) replaced by the requirement that the jump of A be computable fro m 0((n)). These results are best possible in various senses.