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We prove that, for $1 \le p<q<\infty$, the partially ordered set$P(\omega)/{\rm Fin}$ can be embedded into Borel equivalence relations between$\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$. Since there is anantichain of size continuum in $P(\omega)/{\rm Fin}$, therefore there arecontinuum many incomparable Borel equivalence relations between$\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$.

Embeddings of $P(\omega)/{\rm Fin}$ into Borel Equivalence Relations between $\ell_p$ and $\ell_q$