 Kahn-Kalai-Linial theorem

For a Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ under the uniform probability measure, some coordinate has influence of a variable at least $c\mathrm{Var}(f)\log (n)/n$, where $c>0$ is an absolute constant and $I_i(f)$ is the probability that flipping coordinate $i$ changes $f$. For variance bounded below, this is order $\log (n)/n$. The tribes function shows that this order cannot be improved for general balanced Boolean functions.