  The abundance of galactic systems at high redshifts can impose a strong
constraint on the cold+hot dark matter (CDM+HDM) models. The hot component
reduces the excessive small-scale power in the COBE-normalized CDM model but
also delays the epoch of galaxy formation. We present results from the first
numerical simulations that have enough dynamic range to address accurately the
issue of high-redshift halo abundances in CDM+HDM models. Equivalent
high-resolution particle-particle/particle-mesh $N$-body simulations are
performed for spatially flat models with $\Omega_\nu =0.3$ and 0.2 (with
$H_0=50$ km s$^{-1}$ Mpc$^{-1}$ and $\Omega_b=0.05$). We study the constraints
placed on the models by the high-redshift quasar space density and by the mass
fraction in neutral dense gas associated with damped Ly$\alpha$ systems. We
find that even with optimistic assumptions, the much-studied $\Omega_\nu=0.3$
model does not produce enough massive halos to account for the observed
abundance of quasars at $z>4$. The model passes this test if $\Omega_\nu$ is
decreased to 0.2. Both models do not produce enough high column-density halos
to account for the amount of gas in damped Ly$\alpha$ systems at $z\go 3$: the
$\Omega_\nu=0.3$ model falls short by a factor $\sim$80; the $\Omega_\nu=0.2$
model by a factor $\sim$3. We conclude that only CDM+HDM models with
$\Omega_\nu\lo 0.2$ can match observations at high redshift, implying an upper
bound of 4.7 eV on the most massive light neutrino (presumably the $\tau$).

Astrophysics;High Energy Physics - Phenomenology