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We introduce a new invariant of Borel reducibility, namely the notion ofthickness; this associates to every sentence $\Phi$ of $\mathcal{L}_{\omega_1\omega}$ and to every cardinal $\lambda$, the thickness $\tau(\Phi, \lambda)$of $\Phi$ at $\lambda$. As applications, we show that all the Friedman-Stanleyjumps of torsion abelian groups are non-Borel complete. We also show that underthe existence of large cardinals, if $\Phi$ is a sentence of$\mathcal{L}_{\omega_1 \omega}$ with the Schr\"{o}der-Bernstein property (thatis, whenever two countable models of $\Phi$ are biembeddable, then they areisomorphic), then $\Phi$ is not Borel complete.

Borel Complexity and the Schr\"oder-Bernstein Property