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The paper provides a simpler method for proving a delicate inequality thatwas used by Achlioptis and Naor to establish asymptotic concentration forchromatic numbers of Erdos-Renyi random graphs. The simplifications come fromtwo new ideas. The first involves a sharpened form of a piece of statisticalfolklore regarding goodness-of-fit tests for two-way tables of Poisson countsunder linear conditioning constraints. The second idea takes the form of a newinequality that controls the extreme tails of the distribution of a quadraticform in independent Poissons random variables.

Conditioned Poisson distributions and the concentration of chromatic numbers