  We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
  The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union.
