Embeddings of $P(\omega )/\mathrm{Fin}$ into Borel Equivalence Relations between ${\ell }_p$ and ${\ell }_q$

Abstract

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

arXiv comment21 pages