Conditioned Poisson distributions and the concentration of chromatic numbers

The paper provides a simpler method for proving a delicate inequality that was used by Achlioptis and Naor to establish asymptotic concentration for chromatic numbers of Erdös-Rényi random graphs. The simplifications come from two new ideas. The first involves a sharpened form of a piece of statistical folklore regarding goodness-of-fit tests for two-way tables of Poisson counts under linear conditioning constraints. The second idea takes the form of a new inequality that controls the extreme tails of the distribution of a quadratic form in independent Poissons random variables.