  We study the relationship between randomness and effective bi-immunity.
Greenberg and Miller have shown that for any oracle X, there are arbitrarily
slow-growing DNR functions relative to X that compute no ML random set. We show
that the same holds when ML randomness is replaced with effective bi-immunity.
It follows that there are sequences of effective Hausdorff dimension 1 that
compute no effectively bi-immune set.
  We also establish an important difference between the two properties. The
class Low(MLR, EBI) of oracles relative to which every ML random is effectively
bi-immune contains the jump-traceable sets, and is therefore of cardinality
continuum.
