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A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, whenevery sequence of pairwise disjoint normalized elements in E has a subsequenceequivalent to the unit vector basis of l_p. Employing methods frominterpolation theory, we clarify which rearrangement invariant (r.i.) spaces on[0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and anyincreasing concave function f on [0,1], which is not equivalent neither 1 nort, there exists a p-disjointly homogeneous r.i. space with the fundamentalfunction f. Moreover, in the class of all interpolation r.i. spaces withrespect to the Banach couple of Lorentz and Marcinkiewicz spaces with the samefundamental function, dilation indices of which are non-trivial, for every 1<p<infty, there is only a unique p-disjointly homogeneous space.

Disjointly homogeneous rearrangement invariant spaces via interpolation