Symmetric finite representability of $$\ell ^p$$-spaces in rearrangement invariant spaces on [0, 1]

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set all $$p\in [1,\infty ]$$ such that $$\ell ^p$$ is finitely represented in way unit basis vectors ( $$c_0$$ if $$p=\infty $$ ) correspond to pairwise disjoint and equimeasurable functions. This can be treated as follow up paper by first-named author related spaces $$(0,\infty )$$ .