Six Standard Deviations Suffice

With χ : Ω → {−1,+1} random, A ∈ A, χ(A) has zero mean and standard deviation at most √ n. If |χ(A)| > 6√n then χ(A) is at least six standard deviations off the mean. The probability of this occurring is very small but a fixed positive constant and the number of sets A is going to infinity. In fact, a random χ almost always will not work. The specific constant 6 (actually 5.32) was the result of detailed calculations that could certainly be further improved and will not concern us here. Rather we show Theorem 13.2.1 with some constant K replacing 6. The initial argument (found in earlier editions of this work) did not yield an efficient algorithm for finding the desired coloring χ. Indeed, for many years the second author conjectured that no such algorithm would exist. Bansal (2010) gave the first algorithmic argument for Theorem 13.2.1. Here we follow the approach of Lovett and Meka (2012). Their argument is a virtual cornucopia of modern probabilistic methods, we give the basic ideas and leave many of the details to the exercises. We begin by generalizing the problem to vectors.