  A series of examples of toric Sasaki-Einstein 5-manifolds is constructed.
These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein
5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces
examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for
k odd. Toric geometry is used to construct examples of positive Ricci curvature
toric Sasaki structures on non-spin 5-manifolds. Then the join construction is
used to construct infinitely many quasi-regular toric Sasaki-Einstein manifolds
with arbitrarily high second Betti number in every odd dimesion >3.

Differential Geometry