For various lattice gas models with nearest neighbour exclusion
(and, in one case, second-nearest neighbour exclusion as well),
we obtain lower bounds on $m$, the average number of particles on
the non-excluded lattice sites closest to a given particle.
They are all of the form
$$
m/m_{cp} \ge 1 - \hbox{~const.}(N_{cp}/N - 1)
$$
where $N$ is the number of occupied sites,
$m_{cp}$ is the coordination number, snd $N_{cp}$
is the value of $N$ at close packing. An analogous
result exists for hard disks in the plane.
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