A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which admit Riemannian metrics with $\delta$-pinched sectional curvature (i.e. $\delta\le K ≤ 1$).

Are there any known asymptotics on this number as $\delta\to 0$?

A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which admit Riemannian metrics with $\delta$-pinched sectional curvature (i.e. $\delta\le K ≤ 1$).

Are there any known (or conjectured) asymptotics on this number as $\delta\to 0$?