  The one-band Hubbard model at half-filling on a truncated tetrahedron
(C$_{12}$), and on the C$_{60}$ molecule is studied. Within the Hartree-Fock
approximation, we find a magnetic-like instability of the `Fermi sea' towards a
spin-ordered phase for an intermediate value of the coupling ${\rm
(U/t)_c}\approx 2.6$. The ordered phase presents a spin arrangement similar to
that of the classical Heisenberg model defined on the same clusters. We study
the linear excitations around the Hartree-Fock ground states using the
random-phase approximation. On a finite cluster, we expect that these results
signal the presence of a $rapid$ $crossover$ between a paramagnetic region and
a regime where spin correlations are important. The relationship with the
Heisenberg limit (large ${\rm U/t}$) is discussed. Finally, we comment on
implications of our results for purely repulsive models of superconductivity in
alkali-metal-doped fullerenes.
