Conditioned Poisson distributions and the concentration of chromatic numbers

TL;DR

This paper introduces simplified methods for proving key inequalities related to the concentration of chromatic numbers in random graphs, using new ideas in Poisson distribution analysis.

Contribution

It presents novel techniques involving conditioned Poisson distributions and tail inequalities, simplifying previous complex proofs in graph coloring theory.

Findings

New inequality controlling tails of quadratic forms in Poisson variables

Simplified proof of concentration results for chromatic numbers

Enhanced understanding of Poisson distribution behavior under linear constraints

Abstract

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 Erdos-Renyi 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.