  It is shown that the neutron matter interacting through Argonne V18
pair-potential plus modern variants of Urbana or Illinois three-body forces is
unstable. For the energy of $N$ neutrons $E(N)$, which interact through these
forces, we prove mathematically that $E(N) = -cN^3 + \mathcal{O}(N^{8/3})$,
where $c>0$ is a constant. This means that: (i) the energy per particle and
neutron density diverge rapidly for large neutron numbers; (ii) bound states of
$N$ neutrons exist for $N$ large enough. The neutron matter collapse is
possible due to the form of the repulsive core in three-body forces, which
vanishes when three nucleons occupy the same site in space. The old variant of
the forces Urbana VI, where the phenomenological repulsive core does not vanish
at the origin, resolves this problem. We prove that to prevent the collapse one
should add a repulsive term to the Urbana IX potential, which should be larger
than 50 MeV when 3 nucleons occupy the same spatial position.
