  A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when
every sequence of pairwise disjoint normalized elements in E has a subsequence
equivalent to the unit vector basis of l_p. Employing methods from
interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on
[0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any
increasing concave function f on [0,1], which is not equivalent neither 1 nor
t, there exists a p-disjointly homogeneous r.i. space with the fundamental
function f. Moreover, in the class of all interpolation r.i. spaces with
respect to the Banach couple of Lorentz and Marcinkiewicz spaces with the same
fundamental function, dilation indices of which are non-trivial, for every 1<p<
infty, there is only a unique p-disjointly homogeneous space.
